Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x)=(x-k)q(x)+r. ƒ(x)=-3x³+5x-6; k=-1

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Write down the coefficients of the polynomial ƒ(x) = -3x^3 + 0x^2 + 5x - 6. Note that the coefficient of x^2 is 0, so the coefficients are: -3, 0, 5, -6.

Set up the synthetic division using k = -1. Write -1 to the left and the coefficients in a row: -3, 0, 5, -6.

Bring down the first coefficient (-3) as it is. Then multiply it by k (-1) and write the result under the next coefficient. Add the column and write the sum below.

Repeat the multiply and add process for each coefficient: multiply the last sum by k (-1), write it under the next coefficient, and add the column.

After completing the synthetic division, the last number you get is the remainder r. The other numbers form the coefficients of the quotient polynomial q(x). Express ƒ(x) as ƒ(x) = (x - k)q(x) + r using these results.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - k. It simplifies the long division process by using only the coefficients of the polynomial and performing arithmetic operations to find the quotient and remainder quickly.

Polynomial Division Algorithm

The polynomial division algorithm states that for any polynomial ƒ(x) divided by (x - k), there exist a quotient polynomial q(x) and a remainder r such that ƒ(x) = (x - k)q(x) + r. This expresses the original polynomial in terms of its divisor, quotient, and remainder.

Evaluating Polynomials at a Point

Evaluating a polynomial at x = k helps determine the remainder when dividing by (x - k). According to the Remainder Theorem, the remainder r is equal to ƒ(k), which can be found by substituting k into the polynomial.

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