Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.
$x^3+6x^2-2x-7;x+1$

Verified step by step guidance

1. Identify the divisor polynomial and rewrite it in the form \( x - c \). Since the divisor is \( x + 1 \), rewrite it as \( x - (-1) \), so \( c = -1 \).
2. Apply the Factor Theorem by evaluating the first polynomial \( f(x) = x^3 + 6x^2 - 2x - 7 \) at \( x = -1 \). This means calculating \( f(-1) \).
3. Set up synthetic division using \( c = -1 \) and the coefficients of the first polynomial: 1 (for \( x^3 \)), 6 (for \( x^2 \)), -2 (for \( x \)), and -7 (constant term).
4. Perform synthetic division step-by-step: bring down the first coefficient, multiply by \( c \), add to the next coefficient, and repeat until all coefficients are processed.
5. Check the remainder from synthetic division. If the remainder is zero, then \( x + 1 \) is a factor of the first polynomial; otherwise, it is not.

Key Concepts

Factor Theorem

The Factor Theorem states that a polynomial f(x) has (x - c) as a factor if and only if f(c) = 0. This means that substituting c into the polynomial yields zero, confirming that (x - c) divides the polynomial exactly without remainder.

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, making it faster to find the quotient and remainder.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. Determining if one polynomial is a factor of another helps simplify expressions and solve polynomial equations by breaking them into simpler components.