Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = 4x⁴ + x² + 17x + 3; k= -3/2

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = 4x^4 + 0x^3 + 1x^2 + 17x + 3. Note that the coefficient of x^3 is 0, so the coefficients are: 4, 0, 1, 17, 3.
2. Set up synthetic division using k = -\frac{3}{2}. Write the coefficients in a row and place k to the left.
3. Bring down the first coefficient (4) as it is. Then multiply this number by k (4 \times -\frac{3}{2}) and write the result under the next coefficient.
4. Add the second coefficient (0) and the number just written, then repeat the multiply and add process for all coefficients.
5. After completing the synthetic division, check the final number (the remainder). If it is zero, then k is a zero of the polynomial. If not, this remainder is the value of ƒ(k).

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, making it faster and less error-prone. This method helps determine the remainder when the polynomial is evaluated at k.

Zeros of a Polynomial

A zero of a polynomial is a value of x that makes the polynomial equal to zero. If substituting k into the polynomial results in zero, then k is a root or zero of the polynomial. Identifying zeros is essential for factoring and solving polynomial equations.

Evaluating Polynomials

Evaluating a polynomial at a specific value k means substituting k into the polynomial and calculating the result. If the result is zero, k is a zero of the polynomial; otherwise, the result is the value of the polynomial at k, denoted as ƒ(k). Synthetic division provides a quick way to find this value.