1. Suppose f(x) = x^4-2x^3+ax^2+x+3. If f(3) = 2, then what is a? 2. Let f, g, and h be polynomials such that h(x) = f(x) * g(x). If the constant term of f(x) is -4 and the constant term of h(x) is 3, what is g(0)? 3. Suppose the polynomials f and g are both monic polynomials. If the sum f(x) + g(x) is also monic, what can we deduce about the degrees of f and g? 4. If f(x) is a polynomial, is f(x^2) also a polynomial 5. Consider the polynomial function g(x) = x^4-3x^2+9 a. What must be true of a polynomial function f(x) if f(x) and f(-x) are the same polynomial b. What must be true of a polynomial function f(x) if f(x) and -f(-x) are the same polynomial

1. a = -31/9

2. -3/4

3. Different degree polynomials

4. Yes, of a degree 2n

5. a. Even-degree variables

b. Odd- degree variables

Step-by-step explanation:

1. Suppose f(x) = x^4-2x^3+ax^2+x+3. If f(3) = 2, then what is a?

Plugging in 3 for x:

f(3)= 3^4 - 2*3^3 + a*3^2 + 3 + 3= 81 - 54 + 6 + 9a = 33 + 9a and f(3)= 2

9a+33= 29a= -31a = -31/9

2. Let f, g, and h be polynomials such that h(x) = f(x) * g(x). If the constant term of f(x) is -4 and the constant term of h(x) is 3, what is g(0)?

f(0)= -4, h(0)= 3, g(0) = ?h(x)= f(x)*g(x)g(x)= h(x)/f(x)g(0) = h(0)/f(0) = 3/-4= -3/4g(0)= -3/4

3. Suppose the polynomials f and g are both monic polynomials. If the sum f(x) + g(x) is also monic, what can we deduce about the degrees of f and g?

A monic polynomial is a single-variable polynomial in which the leading coefficient is equal to 1.

If the sum of monic polynomials f(x) + g(x) is also monic, then f(x) and g(x) are of different degree and their sum only change the one with the lower degree, leaving the higher degree variable unchanged.

4. If f(x) is a polynomial, is f(x^2) also a polynomial?

If f(x) is a polynomial of degree n, then f(x^2) is a polynomial of degree 2n

5. Consider the polynomial function g(x) = x^4-3x^2+9

a. What must be true of a polynomial function f(x) if f(x) and f(-x) are the same polynomial?

If f(x) and f(-x) are same polynomials, then they have even-degree variables.

b.What must be true of a polynomial function f(x) if f(x) and -f(-x) are the same polynomial?

If f(x) and -f(-x) are the same polynomials, then they have odd-degree variables.

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