Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. $2x^4+5x^3-2x^2+5x+6;x+3$

Verified step by step guidance

1. Identify the divisor polynomial and rewrite it in the form \(x - c\). Since the divisor is \(x + 3\), rewrite it as \(x - (-3)\), so \(c = -3\).
2. Apply the Factor Theorem by evaluating the first polynomial at \(x = -3\). This means substituting \(-3\) into \(2x^4 + 5x^3 - 2x^2 + 5x + 6\) and calculating the result.
3. If the result from step 2 is zero, then \(x + 3\) is a factor of the polynomial. If not, it is not a factor.
4. To confirm, perform synthetic division of the first polynomial by \(x + 3\) using \(c = -3\). Set up the synthetic division with the coefficients of the polynomial: 2, 5, -2, 5, 6.
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 be the last value obtained. If the remainder is zero, \(x + 3\) is a factor; otherwise, it is not.

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 + 3 is a factor, substitute -3 into the polynomial and see if the result is zero. If it is, then x + 3 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. If the remainder is zero, 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.