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)=5x³-3x²+2x-6; k=2

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = 5x^3 - 3x^2 + 2x - 6. These are 5, -3, 2, and -6.
2. Set up the synthetic division by writing the value of k = 2 to the left, and the coefficients in a row to the right: 5, -3, 2, -6.
3. Bring down the first coefficient (5) as it is. Then multiply this number by k (2) and write the result under the next coefficient: 5 × 2 = 10.
4. Add the second coefficient (-3) and the number just written (10): -3 + 10 = 7. Repeat the multiply and add process: multiply 7 by 2, write the result under the next coefficient, then add.
5. Continue this process until all coefficients have been used. The last number you get is the remainder r. The other numbers form the coefficients of the quotient polynomial q(x). Finally, express ƒ(x) as ƒ(x) = (x - 2)q(x) + r.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial 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 yields the quotient and remainder.

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. The remainder is a constant because the divisor is linear, and this form helps in understanding factorization and roots.

Evaluating Remainder Using Remainder Theorem

The Remainder Theorem states that the remainder when a polynomial ƒ(x) is divided by (x - k) is equal to ƒ(k). This provides a quick way to find the remainder without completing the entire division, and it confirms the result obtained from synthetic division.