4.1 Polynomials (in class)

Which of the following expressions is/are polynomial(s)?

If [math](x^2+3x+7)(4x^2-5x-9)=ax^4+bx^3+cx^2+dx+e[/math] where [math]a, b, c, d[/math], and [math]e[/math] are constants, evaluate the following values.

(a) [math]a+e[/math]

(b) [math]a+b+c+d+e[/math]

(c) [math]a-b+c-d+e[/math]

Simplify the following polynomials

(a) [math](x^3+2x^2+7)-2(x^3-6x^2+x+1)[/math]

(b) [math]x(x-1)^3[/math]

Let [math]f(x)=(-a-1)x^2+ax+1[/math] and [math]g(x)=2bx^2-(2+b)x+3[/math] be polynomials where [math]a, b[/math] are real numbers. Suppose [math]\deg(f(x)+g(x))=0[/math]. Find [math]a[/math] and [math]b[/math]

Let [math]f(x)=x^4-4x^2+8x-6[/math] and [math]g(x)=(2x-1)^2[/math]. Find the leading coefficients of following functions:

(a) [math]f(x)\cdot g(x)[/math]

(b) [math]f(g(x))[/math]

(c) [math]3f(x)+2g(x)[/math]

Let [math]f(x)=(x^3+2x^2-3x+4)(x^4-4x^3+3x^2-6x+1)[/math]. Find the coefficient of [math]x^2[/math].