Question

Use the remainder theorem to find

p(-3) for p(x) = 2x^4 + 4x^3 - 4x^2+5

Specifically give the quotient and the remainder for the associated division and the value of p(-3)

Answer 1

Using the remainder theorem, substituting -3 into the polynomial p(x) = 2x^4 + 4x^3 - 4x^2 + 5 gives us p(-3) = 23, which is the remainder when dividing the polynomial by (x + 3).

Step-by-step explanation:

The student is asking to use the remainder theorem to evaluate a polynomial at a given value and to find the quotient and the remainder when dividing the polynomial by a binomial of the form (x - k), where k is the value at which the polynomial is being evaluated.

Specifically, the student needs to find p(-3) for the polynomial p(x) = 2x^4 + 4x^3 - 4x^2 + 5.

Using the remainder theorem, we know that the remainder when p(x) is divided by (x - (-3)), which simplifies to (x + 3), is equal to p(-3). We evaluate p(-3) by substituting -3 into the polynomial:

• p(-3) = 2(-3)^4 + 4(-3)^3 - 4(-3)^2 + 5
• = 2(81) + 4(-27) - 4(9) + 5
• = 162 - 108 - 36 + 5
• = 23

Therefore, p(-3) = 23. This is the remainder, and it means that when dividing the polynomial by (x + 3), the quotient's exact form isn't necessary for the remainder theorem, but can be found through polynomial long division if required.