Answer:
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
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
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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= -3/4
- g(0)= -3/4
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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.
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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
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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.