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math help

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Let f(x) be a quadratic polynomial such that f(-4) = -22, f(-1)=2, and f(2)=-1. Let g(x) = f(x)^16. Find the sum of the coefficients of the terms in g(x) that have even degree. (For example, the sum of the coefficients of the terms in -7x^3 + 4x^2 + 10x - 5 that have even degree is (4) + (-5) = -1.)

Mar 29, 2020

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Mar 29, 2020
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This is just to find the polynomial. I believe you need to use Pascal Triangle to find the rest.

Consider:

Standard form:

ax2 + bx + c = 0

(1)    a(-4)2 + b(-4) + c = -22

(2)    a(-1)2 + b(-1) + c = 2

(3)    a(2)2 + b(2) + c = -1

Simplify:

(1)    16a -4b + c = -22

(2)    a - b + c = 2

(3)    4a + 2b + c = -1

These equations are solvable because there are 3 variables with 3 equations.

Solving:

(1) - (2) = 15a -3b = -24

(2) - (3) = -3a - 3b = 3

Subtract the two resulting equations

18a = -27

a = (-3/2)

From here, it is basic algebra to find the rest of the variables.
a=-(3/2) b=(1/2), c=4

Our quadratic polynomial is:

$$-\frac{3}{2}x^2+\frac{1}{2}x+4=0$$

Mar 29, 2020