Use synthetic division to divide f(x)=x³−4x²+x+6 by x+1. Use the result to find all zeros of f.

Verified step by step guidance

1. Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 1 \), rewrite it as \( x - (-1) \), so \( c = -1 \).
2. Set up synthetic division by writing the coefficients of \( f(x) = x^3 - 4x^2 + x + 6 \) in order: \( 1, -4, 1, 6 \).
3. Perform synthetic division using \( c = -1 \): bring down the first coefficient, multiply by \( c \), add to the next coefficient, and repeat this process for all coefficients.
4. Write the quotient polynomial from the synthetic division result. The degree of the quotient will be one less than the original polynomial, so it will be a quadratic.
5. Use the quotient polynomial to find the remaining zeros of \( f(x) \) by solving the quadratic equation (either by factoring, completing the square, or using the quadratic formula). Remember to include \( x = -1 \) as a zero from the divisor.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form x - c. It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone. This method helps find the quotient and remainder efficiently.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by x - c, the remainder is equal to f(c). If the remainder is zero, then x = c is a root (zero) of the polynomial. This theorem helps verify if a candidate value is a zero of the polynomial.

Finding Zeros of a Polynomial

Finding zeros involves solving f(x) = 0. After dividing the polynomial, the quotient can be factored or solved using other methods to find additional zeros. Identifying all zeros is essential for understanding the polynomial's behavior and graph.