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) = x³ +7x² + 10x; k=0

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = x^3 + 7x^2 + 10x. Since the polynomial is missing a constant term, include 0 for that term. The coefficients are: 1 (for x^3), 7 (for x^2), 10 (for x), and 0 (constant term).
2. Set up the synthetic division using k = 0. Write 0 to the left and the coefficients in a row: 1, 7, 10, 0.
3. Bring down the first coefficient (1) directly below the line.
4. Multiply the number just written below the line by k (which is 0), and write the result under the next coefficient. Then add the column. Repeat this process for each coefficient.
5. After completing the synthetic division, examine the final number (the remainder). If the remainder is 0, then k = 0 is a zero of the polynomial. If not, the remainder is the value of ƒ(k).

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, making it faster and less error-prone. This method helps determine if k is a root by checking if the remainder is zero.

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. This process helps verify if k is a zero by checking if the polynomial equals zero or by finding the value of ƒ(k) when k is not a zero.