Express f(x) in the form f(x)=(x-k) q(x)+r for the given value of k. f(x)=2 x^4+x^3-15 x^2+3 x ;

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

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

Key Concepts

Synthetic Division

Synthetic division is a streamlined process for dividing a polynomial by a linear factor of the form (x - k). It simplifies the division process by reducing the number of operations compared to long division. The method involves using the coefficients of the polynomial and a systematic procedure to obtain the coefficients of the quotient and the remainder, making it particularly efficient for computations with linear divisors.

Remainder Theorem

The remainder theorem is a special case of the polynomial division algorithm applied when dividing by a linear factor of the form (x - k). It states that the remainder of the division of f(x) by (x - k) is equal to f(k). This theorem provides a quick way to determine the remainder without performing long division, as it simply involves evaluating the polynomial at the value k.

Polynomial Division Algorithm

The polynomial division algorithm states that for any two polynomials f(x) and d(x) (with d(x) not being the zero polynomial), there exist unique polynomials q(x) (the quotient) and r(x) (the remainder) such that f(x) = d(x)q(x) + r(x), where the degree of r(x) is less than the degree of d(x). In problems involving division by a linear polynomial, such as (x - k), this guarantees that the remainder is a constant term.

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