Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. 4x²+2x+54; x-4

Verified step by step guidance

1. Identify the polynomials involved: the first polynomial is \$4x^2 + 2x + 54\( and the second polynomial is \)x - 4$.
2. Use the Factor Theorem, which states that if \(x - c\) is a factor of a polynomial, then the polynomial evaluated at \(x = c\) equals zero. Here, set \(x = 4\) because the factor is \(x - 4\).
3. Evaluate the first polynomial at \(x = 4\) by substituting 4 into \$4x^2 + 2x + 54\(, which means calculating \)4(4)^2 + 2(4) + 54$.
4. Perform synthetic division by dividing the first polynomial by \(x - 4\). Set up synthetic division with 4 as the divisor and the coefficients of the first polynomial: 4, 2, and 54.
5. Analyze the remainder from synthetic division: if the remainder is zero, then \(x - 4\) is a factor of the first polynomial; 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. To use it, substitute c into the polynomial; if the result is zero, then (x - c) divides the polynomial exactly.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - c). It simplifies the 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.