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) = 3x⁴ + 4x³ - 10x² + 15; k = -1

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = 3x^4 + 4x^3 - 10x^2 + 0x + 15. Note that the coefficient of x is 0, so the coefficients are: 3, 4, -10, 0, 15.
2. Set up synthetic division using k = -1. Write -1 to the left and the coefficients 3, 4, -10, 0, 15 in a row to the right.
3. Bring down the first coefficient (3) as it is. Then multiply this number by k (-1) and write the result under the next coefficient. Add the column and write the sum below.
4. Repeat the multiply and add process for each coefficient: multiply the last sum by -1, write it under the next coefficient, then add down the column. Continue until all coefficients have been processed.
5. The final row of numbers (except the last one) represents the coefficients of the quotient polynomial q(x). The last number is the remainder r. Express ƒ(x) as ƒ(x) = (x - (-1)) q(x) + r, or ƒ(x) = (x + 1) 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 the value k, making calculations faster and less error-prone.

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.

Polynomial Coefficients and Degree

Understanding the degree of the polynomial and its coefficients is essential for setting up synthetic division correctly. The degree indicates the number of terms, and each coefficient must be placed in order, including zeros for missing terms, to ensure accurate division.