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Consider polynomials F(x) = 4x^3+(a+1)x^2+x-5b and G(x) = 4x^3-ax^2+bx-2,
where a and b are constants.
When both F(x) and G(x) are divided by x-1, the remainders are 3 and -3, respectively.
a) Find the values of a and b.
b) Solve the equation F(x)-G(x) = -2.
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Step by step solution
(a) We are given that F(x) = 4x^3+(a+1)x^2+x-5b gives the remainder 3 when divided by (x-1).
According to the Remainder theorem, it means that F(1) = 3.
Calculate F(1) by substituting x= 1 into the formula for F(x)
F(1) = 4*1 + (a+1)*1 + 1 - 5b = 4 + a+1 + 1 - 5b = a - 5b + 6.
Hence, we have THIS equation for "a" and "b"
a - 5b + 6 = 3, or a - 5b = -3. (1)
(b) Next, we are given that G(x) = 4x^3-ax^2+bx-2 gives the remainder -3 when divided by (x-1).
According to the Remainder theorem, it means that G(1) = -3.
Calculate G(1) by substituting x= 1 into the formula for G(x)
G(1) = 4*1 - a*1 + b*1 - 2 = 4 - a + b - 2 = -a + b + 2.
Hence, we have THIS equation for "a" and "b"
-a + b + 2 = -3, or -a + b = -5. (2)
(c) Thus we have two equations to find "a" and "b"
a - 5b = -3 (1)
-a + b = -5 (2)
To solve, add the equations
-4b = -8 ===> b = (-8)/(-4) = 2.
Then from equation (1),
a = -3 + 5b = -3 + 5*2 = -3 + 10 = 7.
+--------- ANSWER------------+
| Thus a = 7; b = 2. |
+----------------------------+
First part is complete.
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The second part, after subtracting polynomials, gives a quadratic polynomial .
Working with it is simple arithmetic, so I leave this part on you.