+96 1)
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.)
+33661 Question asked and answered here: https://web2.0calc.com/questions/i-m-really-stumped-polynomials
+658
Consider:
Standard form:
ax2 + bx + c = 0
Use your information:
(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:
\(-\frac{3}{2}x^2+\frac{1}{2}x+4=0\)
