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

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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 $.

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) $.

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).

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.x3+6x2−2x−7;x+1x^3+6x^2-2x-7; x+1