Question 149605
# 1


Start with the given zeros


{{{x=1}}} and {{{x=-8}}}



Get all terms to the left side in each case (ie subtract 1 from both sides in the first equation and add 8 to both sides in the second equation)



{{{x-1=0}}} and {{{x+8=0}}}



Now use the zero product property in reverse to join the factors.



{{{(x-1)(x+8)=0}}}



FOIL and multiply



{{{x^2+7x-8=0}}}



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Answer:


So the polynomial with zeros of  {{{x=1}}} and {{{x=-8}}} is


{{{x^2+7*x-8}}}




Notice how if we graph {{{y=x^2+7*x-8}}}, we can see that the polynomial has roots of {{{x=1}}} and {{{x=-8}}}


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+7*x-8 ) }}} Graph of {{{y=x^2+7*x-8}}} with roots of {{{x=1}}} and {{{x=-8}}}




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# 2


Remember the quadratic formula is {{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}}



Where the discriminant is {{{D=b^2-4ac}}}. So the quadratic formula could also look like {{{x = (-b +- sqrt( D ))/(2a)}}}



If D<0, then we'll be taking the square root of a negative number (which we cannot do). So this results in 2 complex solutions (ie no real solutions).




For example, let's find the discriminant for {{{y=x^2+2x+5}}}



From {{{x^2+2x+5}}} we can see that {{{a=1}}}, {{{b=2}}}, and {{{c=5}}}



{{{D=b^2-4ac}}} Start with the discriminant formula



{{{D=(2)^2-4(1)(5)}}} Plug in {{{a=1}}}, {{{b=2}}}, and {{{c=5}}}



{{{D=4-4(1)(5)}}} Square {{{2}}} to get {{{4}}}



{{{D=4-20}}} Multiply {{{4(1)(5)}}} to get {{{(4)(5)=20}}}



{{{D=-16}}} Subtract {{{20}}} from {{{4}}} to get {{{-16}}}



Since the discriminant is less than zero, this means that there are two complex solutions (or no real solutions)



Now let's use the quadratic formula to find the solutions of {{{y=x^2+2x+5}}}



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(2) +- sqrt( (2)^2-4(1)(5) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=2}}}, and {{{c=5}}}



{{{x = (-2 +- sqrt( 4-4(1)(5) ))/(2(1))}}} Square {{{2}}} to get {{{4}}}. 



{{{x = (-2 +- sqrt( 4-20 ))/(2(1))}}} Multiply {{{4(1)(5)}}} to get {{{20}}}



{{{x = (-2 +- sqrt( -16 ))/(2(1))}}} Subtract {{{20}}} from {{{4}}} to get {{{-16}}}



{{{x = (-2 +- sqrt( -16 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-2 +- 4*i)/(2)}}} Take the square root of {{{-16}}} to get {{{4*i}}}. 



{{{x = (-2 + 4*i)/(2)}}} or {{{x = (-2 - 4*i)/(2)}}} Break up the expression. 



{{{x = (-2)/(2) + (4*i)/(2)}}} or {{{x =  (-2)/(2) - (4*i)/(2)}}} Break up the fraction for each case. 



{{{x = -1+2*i}}} or {{{x =  -1-2*i}}} Reduce. 



So our answers are {{{x = -1+2*i}}} or {{{x = -1-2*i}}}



Since our answers are complex (non real), this verifies our original claim.