Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=2 x^3+x^2+x-8 ; k=-1

Name: Problem 19, Chapter 3: Precalculus Student Solutions Manual 5th
Uploaded: 2020-08-04T15:26:52-08:00
Duration: 3 min 1 s
Channel: Cullen Miller
Description: Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=2 x^3+x^2+x-8 ; k=-1

step 1 Hello, hope you're doing well. So we have our function F of X here and our K value here. So sayi

Express f(x) in the form f(x)=(x-k) q(x)+r for the given...

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Key Concepts

Polynomial Division

Polynomial division is a process analogous to long division with numbers, but applied to polynomials. It involves dividing a polynomial by another polynomial to obtain a quotient and a remainder, ensuring that the degree of the remainder is less than that of the divisor.

Division Algorithm for Polynomials

The division algorithm for polynomials states that any polynomial f(x) can be written in the form f(x) = (divisor) × (quotient) + remainder. This representation is fundamental in understanding how polynomials can be decomposed, and it applies when dividing by any non-zero polynomial, particularly when the divisor is linear.

Synthetic Division

Synthetic division is a simplified method of performing polynomial division when dividing by a linear binomial. It streamlines the calculation by using only the coefficients of f(x), making the process faster and less prone to error compared to standard long division, especially in cases where the divisor is of the form (x - c).

Remainder Theorem

The remainder theorem is a principle in algebra that links polynomial division and function evaluation. It states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This theorem is particularly useful for quickly determining the remainder without completing a full division process.

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