Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.
$5 x^{4} + 16 x^{3} - 15 x^{2} + 8 x + 16; x + 4$

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Identify the divisor polynomial and rewrite it in the form $x - c$. Since the divisor is $x + 4$, rewrite it as $x - (-4)$, so $c = -4$.

Apply the Factor Theorem by evaluating the first polynomial at $x = -4$. This means substituting $-4$ into $5x^4 + 16x^3 - 15x^2 + 8x + 16$ and calculating the result.

If the result from step 2 is zero, then $x + 4$ is a factor of the polynomial. If not, it is not a factor.

To confirm, perform synthetic division of the polynomial $5x^4 + 16x^3 - 15x^2 + 8x + 16$ by $x + 4$ using $c = -4$. Set up the synthetic division with coefficients [5, 16, -15, 8, 16].

Carry out the 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. The remainder will tell you if $x + 4$ is a factor (remainder zero) or not (non-zero remainder).

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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 a factor (x - c) if and only if f(c) = 0. To check if a binomial like x + 4 is a factor, substitute -4 into the polynomial and see if the result is zero. If it is, then x + 4 divides the polynomial exactly.

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. The remainder helps verify if the divisor is a factor.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. Identifying factors helps simplify expressions and solve polynomial equations. Using the Factor Theorem and synthetic division together aids in breaking down complex polynomials into simpler components.

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.5x4+16x3−15x2+8x+16;x+45x^4+16x^3-15x^2+8x+16; x+4