Questions: Consider the following polynomial functions. f(x) = x^3 - 16x g(x) = 16x^2 - 8x^3 + x^4 h(x) = x^4 - 8x^3 + 16x^2 + 4 a. For each function, (1) identify its degree, (2) sketch its graph, and (3) find its zeroes. b. Find expressions in standard polynomial form for f(x) + g(x) and f(x) - g(x). c. Find the degrees of these polynomials. i. g(x) - h(x) ii. h(x) - f(x)

Solution Steps

• The degree of \( f(x) = x^3 - 16x \) is 3.
• The degree of \( g(x) = 16x^2 - 8x^3 + x^4 \) is 4.
• The degree of \( h(x) = x^4 - 8x^3 + 16x^2 + 4 \) is 4.

• For \( f(x) = x^3 - 16x \), factor to get \( x(x^2 - 16) = x(x-4)(x+4) \). The zeroes are \( x = 0, 4, -4 \).
• For \( g(x) = 16x^2 - 8x^3 + x^4 \), factor to get \( x^2(x^2 - 8x + 16) = x^2(x-4)^2 \). The zeroes are \( x = 0, 4 \).
• For \( h(x) = x^4 - 8x^3 + 16x^2 + 4 \), use the quadratic formula or numerical methods to find the zeroes. This polynomial does not factor easily, so the zeroes are complex or irrational.

Final Answer