Lesson OVERVIEW of lessons on Evaluating expressions

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<H2>OVERVIEW of lessons on Evaluating expressions</H2>

My lessons on Evaluating expressions in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-expressions-involving-x%2Binv%28x%29-x2%2Binv%28x2%29-and-x%5E3%2Binv%28x%5E3%29.lesson>HOW TO evaluate expressions involving &nbsp;{{{(x + 1/x)}}}, &nbsp;{{{(x^2+1/x^2)}}}, &nbsp;{{{(x^3 + 1/x^3)}}} &nbsp;and &nbsp;{{{(x^5+1/x^5)}}}</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-lesson-on-evaluating-expressions.lesson>Advanced lesson on evaluating expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-functions-of-roots-of-a-square-equation.lesson>HOW TO evaluate functions of roots of a quadratic equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-functions-of-roots-of-a-cubic-and-quartic-equation.lesson>HOW TO evaluate functions of roots of a cubic and quartic equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Problems-on-Vieta%27s-formulas.lesson>Problems on Vieta's formulas</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-problems-on-Vieta%27s-formulas.lesson>Advanced problems on Vieta's theorem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Miscellaneous-problems-on-Vieta%27s-theorem.lesson>Miscellaneous problems on Vieta's theorem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluating-expressions-that-contain-infinitely-many-square-roots.lesson>Evaluating expressions that contain infinitely many square roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Solving-equations-that-contain-infinitely-many-radicals.lesson>Solving equations that contain infinitely many radicals</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Problems-on-evaluating-in-Geometry.lesson>Problems on evaluating in Geometry</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluating-trigonometric-expressions-.lesson>Evaluating trigonometric expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluate-the-sum-of-the-coefficients-of-a-polynomial.lesson>Evaluate the sum of the coefficients of a polynomial</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Miscellaneous-evaluating-problems.lesson>Miscellaneous evaluating problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-evaluating-problems.lesson>Advanced evaluating problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Lowering-a-degree-method.lesson>Lowering a degree method</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Find-the-number-of-factorable-quadratic-polynomials-of-special-form.lesson>Find the number of factorable quadratic polynomials of special form</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluating-a-function-defined-by-functional-equation.lesson>Evaluating a function defined by functional equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Math-circle-level-problems-on-evaluating-expressions.lesson>Math circle level problems on evaluating expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Math-circle-level-problem-on-finding-polynomial-with-assigned-roots.lesson>Math circle level problems on finding polynomials with prescribed roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Math-Olympiad-level-problem-on-evaluating-of-a-9-degree-polynomial.lesson>Math Olympiad level problem on evaluating a 9-degree polynomial</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Upper-league-problem-on-evaluating-the-sum.lesson>Upper league problem on evaluating the sum</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Finding-coefficients-of-decomposition-of-a-rational-function.lesson>Finding coefficients of decomposition of a rational function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Upper-level-problem-on-evaluating-an-expression-of-polynomial-roots.lesson>Upper level problems on evaluating an expression of polynomial roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/A-truly-miraculous-evaluating-problem.lesson>A truly miraculous evaluating problem with a truly miraculous solution</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Entertainment-problems-on-evaluating-expressions.lesson>Entertainment problems on evaluating expressions</A>

<H3>List of the lessons with short annotations</H3>

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-expressions-involving-x%2Binv%28x%29-x2%2Binv%28x2%29-and-x%5E3%2Binv%28x%5E3%29.lesson>HOW TO evaluate expressions involving &nbsp;{{{(x + 1/x)}}}, &nbsp;{{{(x^2 + 1/x^2)}}}, &nbsp;{{{(x^3 + 1/x^3)}}} &nbsp;and &nbsp;{{{(x^5 + 1/x^5)}}}</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;{{{x}}} + {{{1/x}}} = {{{3}}}, &nbsp;then find &nbsp;{{{x^2}}} + {{{1/x^2}}} &nbsp;and &nbsp;{{{x^3}}} + {{{1/x^3}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;{{{x}}} + {{{1/x}}} = {{{7}}}, &nbsp;then find &nbsp;{{{x^2}}} + {{{1/x^2}}} &nbsp;and &nbsp;{{{x^3}}} + {{{1/x^3}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;{{{x}}} + {{{1/x}}} = {{{7}}}, &nbsp;then find &nbsp;{{{sqrt(x)}}} + {{{1/sqrt(x)}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;If x = {{{3+sqrt(8)}}}, &nbsp;then find &nbsp;{{{(x^6+x^4+x^2+1)/x^3}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;If &nbsp;&nbsp;{{{x}}} + {{{1/x}}} = 5, &nbsp;&nbsp;find &nbsp;&nbsp;{{{x^5}}} + {{{1/x^5}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;If &nbsp;x&nbsp; is the root of the equation &nbsp;{{{x^2 - 3x + 1}}} = 0, &nbsp;find &nbsp;&nbsp;{{{x^5}}} + {{{1/x^5}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;If &nbsp;&nbsp;{{{m}}} - {{{1/m}}} = 5, &nbsp;find &nbsp;&nbsp;{{{m^2}}} - {{{1/m^2}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-lesson-on-evaluating-expressions.lesson>Advanced lesson on evaluating expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;&nbsp;{{{a}}} + {{{1/(3a)}}} = 2, &nbsp;then find the value of &nbsp;&nbsp;{{{9a^2}}} + {{{1/a^2}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;&nbsp;{{{a}}} + {{{1/a}}} = 2, &nbsp;find &nbsp;&nbsp;{{{9a^2}}} + {{{1/a^2}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Given the following equalities &nbsp;&nbsp;g(x) + g(x + 3) = 2x + 5 &nbsp;&nbsp;and &nbsp;&nbsp;g(2) + g(8) = 12, &nbsp;find &nbsp;&nbsp;g(5).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;If &nbsp;{{{(x+(1/x))^2}}} = 7 &nbsp;and &nbsp;x&nbsp; is a positive real number, &nbsp;find the exact value of &nbsp;{{{x^3}}} + {{{1/x^3}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;If &nbsp;{{{x^2}}} + {{{1/x^2}}} = 102, &nbsp;find the value of &nbsp;{{{x^3}}} - {{{1/x^3}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;If &nbsp;&nbsp;x+y=1 &nbsp;&nbsp;and &nbsp;&nbsp;{{{x^3+y^3=19}}}, &nbsp;find the value of &nbsp;&nbsp;{{{x^2+y^2}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;If &nbsp;{{{x^2+x-1}}} = 0, &nbsp;what is the value of &nbsp;{{{x^4+(1/x^4)}}} ?

<A HREF=http://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-functions-of-roots-of-a-square-equation.lesson>HOW TO evaluate functions of roots of a quadratic equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;r &nbsp;and &nbsp;s &nbsp;are the roots of the equation &nbsp;&nbsp;{{{x^2 -8x +6}}} = {{{0}}}, &nbsp;&nbsp;then evaluate these expressions:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;{{{r^2 + s^2}}}; &nbsp;&nbsp;2) &nbsp;{{{r^2 + 3rs +s^2}}}; &nbsp;&nbsp;3) &nbsp;{{{r^3 + s^3}}} &nbsp;&nbsp;and &nbsp;&nbsp;4) &nbsp;{{{1/r}}} + {{{1/s}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;r &nbsp;and &nbsp;s &nbsp;are the roots of the equation &nbsp;&nbsp;{{{x^2 +px +q}}} = {{{0}}}, &nbsp;&nbsp;then evaluate these expressions:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;r &nbsp;and &nbsp;s &nbsp;are the roots of the quadratic equation &nbsp;&nbsp;{{{3x^2 - 4x + 7}}} = {{{0}}}, &nbsp;&nbsp;then find:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;{{{r^2 + s^2}}}, &nbsp;2) &nbsp;{{{r^2 + 3rs + s^2}}}, &nbsp;&nbsp;3) &nbsp;{{{r/s}}} + {{{s/r}}} &nbsp;&nbsp;and &nbsp;&nbsp;4) &nbsp;{{{1/r^2}}} + {{{1/s^2}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-functions-of-roots-of-a-cubic-and-quartic-equation.lesson>HOW TO evaluate functions of roots of a cubic and quartic equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;r, &nbsp;s &nbsp;and &nbsp;t &nbsp;are the roots of the cubic equation &nbsp;&nbsp;{{{x^3-7x+5}}} = {{{0}}}, &nbsp;&nbsp;find:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;r + s + t &nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;rs + rt + st; &nbsp;&nbsp;&nbsp;&nbsp;3) &nbsp;rst; &nbsp;&nbsp;&nbsp;&nbsp;4) &nbsp;{{{r^2 + s^2 + t^2}}};  &nbsp;&nbsp;&nbsp;&nbsp;5) &nbsp;{{{1/r + 1/s + 1/t}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;r, &nbsp;s &nbsp;and &nbsp;t &nbsp;are the roots of the cubic equation &nbsp;&nbsp;{{{2x^3-7x+5}}} = {{{0}}}, &nbsp;&nbsp;find:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;r + s + t &nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;rs + rt + st; &nbsp;&nbsp;&nbsp;&nbsp;3) &nbsp;rst; &nbsp;&nbsp;&nbsp;&nbsp;4) &nbsp;{{{r^2 + s^2 + t^2}}};  &nbsp;&nbsp;&nbsp;&nbsp;5) &nbsp;{{{1/r + 1/s + 1/t}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;r, &nbsp;s, &nbsp;t &nbsp;and &nbsp;u &nbsp;are the roots of the quartic equation &nbsp;&nbsp;{{{x^4-3x^3 + 5x^2 - 7x + 9}}} = {{{0}}}, &nbsp;&nbsp;find:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;r + s + t + u &nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;rs + rt + ru + st + su + tu; &nbsp;&nbsp;&nbsp;&nbsp;3) &nbsp;rst + stu + rtu + rsu; &nbsp;&nbsp;&nbsp;&nbsp;4) &nbsp;rstu; &nbsp;&nbsp;&nbsp;&nbsp;5) &nbsp;{{{r^2 + s^2 + t^2 + u^2}}};  &nbsp;&nbsp;6) &nbsp;{{{1/r + 1/s + 1/t + 1/u}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Problems-on-Vieta%27s-formulas.lesson>Problems on Vieta's formulas</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;x&nbsp; is the root of the equation &nbsp;&nbsp;{{{x^2 - 3x + 1}}} = 0, &nbsp;find &nbsp;&nbsp;{{{x^5}}} + {{{1/x^5}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If the sum of the reciprocals of the roots of the quadratic equation &nbsp;&nbsp;3x^2 + 7x + k = 0 &nbsp;&nbsp;is &nbsp;-7/3, &nbsp;what is &nbsp;k?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;p&nbsp; and &nbsp;q&nbsp; are the roots of the equation &nbsp;{{{2x^2-x-4=0}}}, &nbsp;find the equation whose roots are &nbsp;{{{p-(q/p)}}} &nbsp;and &nbsp;{{{q-(p/q)}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Consider polynomial &nbsp;&nbsp;P(z) = z^5 - 10z^2 + 15z - 6 &nbsp;&nbsp;of complex variable &nbsp;z.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Given that &nbsp;&nbsp;{{{x^2-3x+2}}} &nbsp;&nbsp;is a factor of &nbsp;&nbsp;{{{x^4 + kx^3 - 10x^2 - 20x+24}}}, &nbsp;&nbsp;evaluate

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-problems-on-Vieta%27s-formulas.lesson>Advanced problems on Vieta's theorem</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;One of the roots of the equation &nbsp;{{{2x^2 + 1}}} = {{{h-5x}}}&nbsp; is four times to other root. &nbsp;Find the value oh &nbsp;h.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;The horizontal line &nbsp;y = k &nbsp;intersects the parabola with equation &nbsp;y = 2(x-3)(x-5) &nbsp;at points &nbsp;A &nbsp;and &nbsp;B.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Find the sum of the squares of the roots to the equation &nbsp;(4x^2 - 9)^4 - 10(4x^2 - 9)^2 + 9 = 0.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;If the solution for &nbsp;x&nbsp; of &nbsp;&nbsp;{{{x^2+px+q}}} = {{{0}}} &nbsp;&nbsp;are the cubes of the solutions for &nbsp;&nbsp;{{{x^2+mx+n}}} = {{{0}}}, &nbsp;&nbsp;express &nbsp;p&nbsp; and &nbsp;q&nbsp; in terms of &nbsp;m&nbsp; and &nbsp;n.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Circle T intersects the hyperbola &nbsp;y = {{{1/x}}} &nbsp;at &nbsp;(1,1), &nbsp;(3, {{{1/3}}}), &nbsp;and two other points.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;Let &nbsp;x, &nbsp;y, &nbsp;and &nbsp;z &nbsp;be real numbers such that &nbsp;x + y + z, &nbsp;xy + xz + yz, &nbsp;and &nbsp;xyz &nbsp;are all positive.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;Determine &nbsp;(r + s)(s + t)(t + r), &nbsp;if &nbsp;r, &nbsp;s, &nbsp;and &nbsp;t &nbsp;are the three real roots of the polynomial &nbsp;&nbsp;{{{x^3 + 9x^2 - 9x - 8}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 8. &nbsp;Solve equation &nbsp;&nbsp;2x^3 + 3x^2 + hx + k = 0 &nbsp;&nbsp;and find the values of &nbsp;h &nbsp;and &nbsp;k, &nbsp;given that &nbsp;-3 &nbsp;is the first root and the third root is twice the second.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 9. &nbsp;Find the quadratic equation such that each of its roots is the sum of a root

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 10. &nbsp;Write an equation each of whose roots are &nbsp;2 &nbsp;less than &nbsp;3 &nbsp;times the roots of &nbsp;&nbsp;3x^3 + 10x^2 + 7x - 10 = 0.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Miscellaneous-problems-on-Vieta%27s-theorem.lesson>Miscellaneous problems on Vieta's theorem</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;'a' &nbsp;and &nbsp;'b' &nbsp;be the roots of the quadratic equation &nbsp;&nbsp;5x^2 - 23x - 4 = 0.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute &nbsp;&nbsp;1/a^2 + 1/b^2.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Let &nbsp;'a' &nbsp;and '&nbsp;b' &nbsp;be the solutions to &nbsp;&nbsp;6x^2 - 16x + 7 = 0.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find &nbsp;&nbsp;a^2/b + b^2/a.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Let &nbsp;'a' &nbsp;and &nbsp;'b' &nbsp;be the roots of the quadratic &nbsp;&nbsp;x^2 - 5x + 3 = 0.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;There are integers &nbsp;'b', &nbsp;'c' &nbsp;for which both roots of the polynomial &nbsp;&nbsp;x^2 - x - 3

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Let &nbsp;'p', &nbsp;'q', &nbsp;'r', &nbsp;and &nbsp;'s' &nbsp;be the roots of &nbsp;&nbsp;g(x) = x^4 + 2x^3 + 16x^2 + 20x - 31.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;Let &nbsp;'p', &nbsp;'q', &nbsp;'r', &nbsp;and &nbsp;'s' be the roots of &nbsp;&nbsp;x^4 + 2x^3 + 16x^2 + 20x - 31.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute &nbsp;&nbsp;p^2 + q^2 + r^2 + s^2.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluating-expressions-that-contain-infinitely-many-square-roots.lesson>Evaluating expressions that contain infinitely many square roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Simplfy/evaluate &nbsp;&nbsp;{{{sqrt(5 + sqrt(5 + sqrt(5 + ellipsis)))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Simplfy &nbsp;&nbsp;{{{sqrt(2 + sqrt(2 + sqrt(2 + ellipsis)))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Simplfy &nbsp;&nbsp;{{{sqrt(6 + sqrt(6 + sqrt(6 + ellipsis)))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Simplfy &nbsp;&nbsp;{{{sqrt(12 + sqrt(12 + sqrt(12 + ellipsis)))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Evaluate &nbsp;&nbsp;{{{sqrt(7/3 + sqrt(7/9 + sqrt(7/3 + sqrt(7/9) + ellipsis)))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;Evaluate the value &nbsp;&nbsp;{{{sqrt(5+sqrt(5^2+sqrt(5^4+sqrt(5^8+ellipsis))))}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Solving-equations-that-contain-infinitely-many-radicals.lesson>Solving equations that contain infinitely many radicals</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Solve an equation {{{sqrt(y+sqrt(y+sqrt(y+ellipsis)))}}} = {{{5}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Let &nbsp;&nbsp;a = {{{root(3, a^2 + root(3, a^2 + root(3, a^2 + ellipsis)))}}}. &nbsp;&nbsp;Prove that &nbsp;&nbsp;a = {{{root(7, 4a^4 + root(7, 4a^4 + root(7, 4a^4 + ellipsis)))}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Problems-on-evaluating-in-Geometry.lesson>Problems on evaluating in Geometry</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A cuboid has latent faces and top/bottom faces of areas &nbsp;10 cm^2, &nbsp;7.5 cm^2 &nbsp;and &nbsp;3 cm^2, &nbsp;respectively.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;What is the volume of the solid in cm^3 ?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Consider right rectangular prism. &nbsp;The total surface area of the prism is &nbsp;1 square unit.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluating-trigonometric-expressions-.lesson>Evaluating trigonometric expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;What is the value of &nbsp;cos(2a), &nbsp;if &nbsp;sin(3a) = 2*sin(a)?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;What is the value of &nbsp;cos(2a) , &nbsp;if &nbsp;2*cos(3a)= cos(a)?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If {{{sin^2(pi/9)}}} + {{{sin^2(2pi/9)}}} + {{{sin^2(3pi/9)}}} + {{{sin^2(4pi/9)}}} = 9/4,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Find value of &nbsp;&nbsp;cos(2π/7) + cos(4π/7) + cos(6π/7).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;If &nbsp;&nbsp;sin⁶β + cos⁶β = ¼, &nbsp;&nbsp;find (1/sin⁶β) + (1/cos⁶β).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;If &nbsp;&nbsp;8*cos(a) - 8*sin(a) = 3, &nbsp;&nbsp;find &nbsp;&nbsp;{{{55*tan(a) }}} + {{{55/tan(a)}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluate-the-sum-of-the-coefficients-of-a-polynomial.lesson>Evaluate the sum of the coefficients of a polynomial</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;&nbsp;F(x) = ax^2+bx+c, &nbsp;and &nbsp;&nbsp;F(x+5) = x^2+9x-7, &nbsp;what is the sum of &nbsp;&nbsp;a+b+c?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;A quartic polynomial in the form &nbsp;&nbsp;f(x) = ax^4 + bx^3 + cx^2 + dx + e &nbsp;&nbsp;is such that the coefficient

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the quadratic and linear terms are &nbsp;10 &nbsp;and &nbsp;-18 &nbsp;respectively. &nbsp;Additionally, &nbsp;f(0)= &nbsp;9 &nbsp;and &nbsp;x= 1 &nbsp;is a root. &nbsp;&nbsp;What is the value of &nbsp;(a + b)?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Let &nbsp;f(x) &nbsp;be a quadratic polynomial such that &nbsp;f(-4) = -22, &nbsp;f(-1)=2, &nbsp;and &nbsp;f(2)=-1. &nbsp;Let &nbsp;g(x) = f(x)^{16}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Miscellaneous-evaluating-problems.lesson>Miscellaneous evaluating problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;a-b = 4 &nbsp;and &nbsp;ab+c^2+4 = 0, &nbsp;find a+b.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;&nbsp;(2006-a)(2004-a)=2005, &nbsp;find (2006-a)^2 + (2004-a)^2.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;a-b = -6 &nbsp;and &nbsp;b+c = 9, &nbsp;&nbsp;find the value of &nbsp;&nbsp;{{{3a^2-b^2-2c^2}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Evaluate the following sums

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Express the following product as reduced fraction &nbsp;&nbsp;{{{(1-1/2)(1-1/3)(1-1/4)*ellipsis*(1-1/2008)}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;Find the value of the product &nbsp;&nbsp;{{{(1-1/2^2)(1-1/3^2)(1-1/4^2)*ellipsis*(1-1/999^2)}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;Find the sum of the sequences &nbsp;8, &nbsp;88, &nbsp;888, &nbsp;. . . &nbsp;up to &nbsp;n&nbsp; terms.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 8. &nbsp;Find the sum of all possible &nbsp;4-digit numbers that can be formed using the digits &nbsp;2, &nbsp;4, &nbsp;5, &nbsp;6, &nbsp;7, &nbsp;and &nbsp;8, &nbsp;with no repeated digits.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 9. &nbsp;If &nbsp;&nbsp;x + y + z = 20 &nbsp;and &nbsp;&nbsp;x^2 + y^2 + z^2 = 100, &nbsp;find the value of &nbsp;&nbsp;xy + xz + yz.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 10. &nbsp;If &nbsp;&nbsp;F(x) = ax^2 +bx+c &nbsp;&nbsp;and &nbsp;&nbsp;F(x+5) = x^2+9x-7, &nbsp;&nbsp;find the sum of &nbsp;&nbsp;a+b+c.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 11. &nbsp;For positive numbers &nbsp;a, &nbsp;b, &nbsp;and c, &nbsp;if 2ab = 1, &nbsp;3bc = 2, &nbsp;and &nbsp;4ca = 3, &nbsp;find the value of &nbsp;a + b + c.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 12. &nbsp;All students in &nbsp;Ms. &nbsp;Fay's &nbsp;Spanish class are either going on the &nbsp;Spain trip, &nbsp;the &nbsp;Mexico trip, &nbsp;or both.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(A) &nbsp;&nbsp;26; &nbsp;&nbsp;&nbsp;&nbsp;(B) &nbsp;&nbsp;27; &nbsp;&nbsp;&nbsp;&nbsp;(C) &nbsp;&nbsp;28; &nbsp;&nbsp;&nbsp;&nbsp;(D) &nbsp;&nbsp;29; &nbsp;&nbsp;&nbsp;&nbsp;(E) &nbsp;&nbsp;30.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 13. &nbsp;If &nbsp;&nbsp;{{{2^x}}} = {{{4^y}}} = {{{8^z}}} &nbsp;&nbsp;and &nbsp;&nbsp;{{{1/2x}}} + {{{1/4y}}} + {{{1/8z}}} = {{{22/7}}}, &nbsp;&nbsp;show that &nbsp;&nbsp;x = {{{7/16}}}, &nbsp;y = {{{7/32}}} &nbsp;and &nbsp;z = {{{7/48}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-evaluating-problems.lesson>Advanced evaluating problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;x &nbsp;be a real number such that &nbsp;&nbsp;{{{625^x}}} = 64.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;{{{x^2 + y^2 + z^2}}} = {{{2*(5X - 8Y + 6Z) - 125}}}, &nbsp;where &nbsp;x, &nbsp;y &nbsp;and &nbsp;z &nbsp;are real numbers, &nbsp;find the value of &nbsp;x+y+z.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;&nbsp;f(x) = {{{9^x/(3 + 9^x)}}}, &nbsp;&nbsp;prove that &nbsp;&nbsp;{{{f(1/2016)}}} + {{{f(2/2016)}}} + {{{f(3/2016)}}} + . . . + {{{f(2015/2016)}}} = {{{2015/2}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;If &nbsp;{{{1/x}}} - {{{1/y}}} = {{{1/(x+y)}}}, &nbsp;&nbsp;find &nbsp;&nbsp;{{{y/x}}} - {{{x/y}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;If &nbsp;&nbsp;{{{1/x}}} - {{{1/y}}} = {{{1/(x+y)}}}, &nbsp;&nbsp;find &nbsp;&nbsp;{{{y/x}}} + {{{x/y}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;The lengths of the sides of an equilateral triangle are &nbsp;{{{log(4,a)}}}, &nbsp;{{{log(10,b)}}}, &nbsp;{{{log(25,(a+b))}}},

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;If &nbsp;&nbsp;(1 + a)(1 + b)(a + b) = 1530 &nbsp;&nbsp;and &nbsp;&nbsp;a^3 + b^3 = 1241, &nbsp;&nbsp;find (a + b).

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Lowering-a-degree-method.lesson>Lowering a degree method</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;"a" &nbsp;and &nbsp;"b" &nbsp;are the roots of equation &nbsp;&nbsp;{{{x^2}}} = x+1, &nbsp;find the value of &nbsp;&nbsp;{{{a^5}}} + {{{b^5}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;&nbsp;x^3 + 5x - 10 = 0, &nbsp;&nbsp;then find the value of &nbsp;&nbsp;x^7 + 100x^2 + 25x.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Suppose x is a positive number such that &nbsp;&nbsp;{{{x^2 = 1-x}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Find-the-number-of-factorable-quadratic-polynomials-of-special-form.lesson>Find the number of factorable quadratic polynomials of special form</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the number of positive integers &nbsp;n, &nbsp;&nbsp;1 <= n <= 1000, &nbsp;&nbsp;for which

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Evaluating-a-function-defined-by-functional-equation.lesson>Evaluating a function defined by functional equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The function &nbsp;f &nbsp;satisfies &nbsp;&nbsp;{{{f(sqrt(2x - 1))}}} = {{{1/(2x - 1)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Suppose a function &nbsp;f &nbsp;is such that &nbsp;&nbsp;f(1/x) - 3f(x) = x &nbsp;&nbsp;for every non-zero &nbsp;x. &nbsp;&nbsp;Find &nbsp;f(2).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;If &nbsp;&nbsp;{{{f(3x/(x-4))}}} = {{{x^2 + x + 1}}}, &nbsp;&nbsp;what is the value of &nbsp;f(5)?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;If &nbsp;&nbsp;f(x) = 1/(1 - x) , &nbsp;&nbsp;find &nbsp;&nbsp;(f(f(f(f...f)(sqrt2), &nbsp;(45 times).

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Math-circle-level-problems-on-evaluating-expressions.lesson>Math circle level problems on evaluating expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;&nbsp;f(2a-b) = f(a)*f(b) &nbsp;for all &nbsp;"a" &nbsp;and &nbsp;"b", &nbsp;and the function is never equal to zero,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;find the value of &nbsp;f(5).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;A function &nbsp;f &nbsp;is defined for integers &nbsp;m &nbsp;and &nbsp;n &nbsp;as given:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) Prove that &nbsp;f(x) = f(x-1) + 1001.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b) Find the value of &nbsp;f(9999).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Given that &nbsp;&nbsp;{{{f(x)=5x^2-3x+7}}} &nbsp;&nbsp;and &nbsp;&nbsp;{{{f(g(x))=(5x^4/9)+(17x^2/3)+21}}}, &nbsp;&nbsp;find all possible values

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Five different positive integers added two at a time give the following sums: &nbsp;16, &nbsp;20, &nbsp;22, &nbsp;23, &nbsp;25, &nbsp;28, &nbsp;29, &nbsp;30, &nbsp;34 &nbsp;and &nbsp;37.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the product of the five integers.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;There are &nbsp;5 &nbsp;sacks, &nbsp;and they are weighed &nbsp;2&nbsp; at a time.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Their weights are &nbsp;11, &nbsp;11.2, &nbsp;11.3, &nbsp;11.4, &nbsp;11.5, &nbsp;11.6, &nbsp;11.7, &nbsp;11.8, &nbsp;12 and &nbsp;12.1.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Math-circle-level-problem-on-finding-polynomial-with-assigned-roots.lesson>Math circle level problems on finding polynomials with prescribed roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the polynomial with roots &nbsp;{{{alpha}}}, &nbsp;{{{beta}}} &nbsp;and &nbsp;{{{gamma}}}, &nbsp;if &nbsp;{{{alpha*beta*gamma}}} = 6, &nbsp;{{{alpha + beta + gamma}}} = 5,&nbsp; and &nbsp;{{{alpha^2 + beta^2 + gamma^2}}} =21.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Let the roots of the equation &nbsp;&nbsp;x^3 -2x^2 -3x-7=0 &nbsp;&nbsp;be &nbsp;r, &nbsp;s, &nbsp;and &nbsp;t.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;The roots of the polynomial equation &nbsp;{{{2x^3 - 8x^2 + 3x + 5}}} = 0 &nbsp;are &nbsp;{{{alpha}}}, &nbsp;{{{beta}}} &nbsp;and &nbsp;{{{gamma}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Use this identify

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;to find the polynomial of least degree that has zeroes &nbsp;{{{(tan(pi/24))^2}}}, &nbsp;{{{(tan(7pi/24))^2}}}, &nbsp;{{{(tan(13pi/24))^2}}}, &nbsp;{{{(tan(19pi/24))^2)}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Math-Olympiad-level-problem-on-evaluating-of-a-9-degree-polynomial.lesson>Math Olympiad level problem on evaluating a 9-degree polynomial</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;&nbsp;P(x) = {{{2009x^9}}} + {{{a[1]x^8}}} + . . . + {{{a[9]}}} &nbsp;&nbsp;such that &nbsp;&nbsp;{{{P(1/n)}}} = {{{1/(n^3)}}}, &nbsp;n = 1, 2, . . . , 9.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find &nbsp;&nbsp;{{{P(1/10)}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Upper-league-problem-on-evaluating-the-sum.lesson>Upper league problem on evaluating the sum</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;&nbsp;f(n) = {{{log((n))/log((2006n-n^2))}}}, &nbsp;&nbsp;&nbsp;find &nbsp;&nbsp;f(1) + f(2) + f(3) + . . . + f(2005).

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Finding-coefficients-of-decomposition-of-a-rational-function.lesson>Finding coefficients of decomposition of a rational function</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find &nbsp;A, &nbsp;B &nbsp;and &nbsp;C &nbsp;if &nbsp;&nbsp;{{{A/(x-1)}}} + {{{B/(x-2)}}} + {{{C/(x-3)}}} = {{{(2x^2-6x+6)/(x-1)(x-2)(x-3)}}}

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Upper-level-problem-on-evaluating-an-expression-of-polynomial-roots.lesson>Upper level problems on evaluating an expression of polynomial roots</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;a, &nbsp;b, &nbsp;c &nbsp;&nbsp;(where &nbsp;a, &nbsp;b, &nbsp;c =/= 0) &nbsp;&nbsp;are the roots of the equation &nbsp;&nbsp;{{{x^3 + px^2 + qx + r}}} = 0,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;where &nbsp;p, &nbsp;q &nbsp;and &nbsp;𝑟 (=/= 0) &nbsp;are real numbers, &nbsp;express &nbsp;&nbsp;{{{1/a^3}}} + {{{1/b^3}}} + {{{1/c^3}}} &nbsp;&nbsp;in terms of &nbsp;p, &nbsp;q &nbsp;and &nbsp;r.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;&nbsp;{{{x^5}}} = 1 &nbsp;&nbsp;with x =/= 1, &nbsp;find the value of {{{1/(1+x^2)}}} + {{{1/(1+x^4)}}} + {{{1/(1+x)}}} + {{{1/(1+x^3)}}} .

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/A-truly-miraculous-evaluating-problem.lesson>A truly miraculous evaluating problem with a truly miraculous solution</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Given &nbsp;&nbsp;(x² + 1)(y² + 1) + 25 = 10(x + y), &nbsp;&nbsp;find (x³ + y³)/220.

<A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Entertainment-problems-on-evaluating-expressions.lesson>Entertainment problems on evaluating expressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;&nbsp;{{{x^2}}} - {{{x}}} + {{{1}}} = {{{0}}}, &nbsp;&nbsp;find &nbsp;{{{x^2020}}} + {{{x^1010}}} - {{{1}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;If &nbsp;&nbsp;2^a = 3, &nbsp;3^b = 2, &nbsp;&nbsp;find &nbsp;&nbsp;1/(a+1) + 1/(b+1).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Let &nbsp;p, &nbsp;q, &nbsp;r, &nbsp;and &nbsp;s &nbsp;be the roots of &nbsp;&nbsp;g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute &nbsp;&nbsp;1/p + 1/q + 1/r + 1/s.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.