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

Verified step by step guidance

1. Identify the divisor polynomial and rewrite it in the form \(x - c\). Since the divisor is \(x + 4\), rewrite it as \(x - (-4)\), so \(c = -4\).
2. 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.
3. If the result from step 2 is zero, then \(x + 4\) is a factor of the polynomial. If not, it is not a factor.
4. 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].
5. 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).

Key Concepts

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.