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) = 2x³ + 3x² - 16x+10; k = -4

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = 2x^3 + 3x^2 - 16x + 10. These are: 2, 3, -16, and 10.
2. Set up synthetic division using k = -4. Write -4 to the left and the coefficients in a row: 2, 3, -16, 10.
3. Bring down the first coefficient (2) as it is. Then multiply this number by k (-4) and write the result under the next coefficient.
4. Add the numbers in the second column, write the sum below, then repeat the multiply and add process for each column until you reach the last coefficient.
5. The numbers you get at the bottom (except the last one) are the coefficients of the quotient polynomial q(x). The last number is the remainder r. Express ƒ(x) as ƒ(x) = (x - (-4)) q(x) + r, or ƒ(x) = (x + 4) q(x) + r.

Key Concepts

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 in a tabular form. This method quickly provides the quotient and remainder.

Polynomial Division and Remainder Theorem

When dividing a polynomial ƒ(x) by (x - k), the result can be expressed as ƒ(x) = (x - k)q(x) + r, where q(x) is the quotient polynomial and r is the remainder. The Remainder Theorem states that the remainder r equals ƒ(k), the value of the polynomial evaluated at k.

Expressing the Polynomial in Division Form

After performing synthetic division, the original polynomial is rewritten as the product of the divisor (x - k) and the quotient q(x), plus the remainder r. This form clearly shows the relationship between the original polynomial and its factors, aiding in factorization and root analysis.