Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 4x^2+2x+54; x-4

Verified step by step guidance

1. Identify the divisor from the second polynomial, which is \(x - 4\). The root to test using the Factor Theorem is \(x = 4\).
2. Set up synthetic division using the coefficients of the first polynomial \(4x^2 + 2x + 54\), which are \([4, 2, 54]\).
3. Perform synthetic division by bringing down the leading coefficient (4) and multiply it by the root (4), then add to the next coefficient.
4. Continue the synthetic division process: multiply the result by the root and add to the next coefficient until all coefficients are processed.
5. Check the remainder: if the remainder is 0, then \(x - 4\) is a factor of \(4x^2 + 2x + 54\). If not, it is not a factor.

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. This theorem is essential for determining whether a given polynomial is a factor of another polynomial. By evaluating the polynomial at specific values, we can identify roots and thus confirm factors.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for quicker calculations when dividing a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions to find the quotient and remainder, which helps in verifying factors.

Polynomials

Polynomials are algebraic expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Understanding the structure of polynomials, including their degree and coefficients, is crucial for applying the Factor Theorem and synthetic division effectively in problem-solving.