Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. x³-5x²+3x+1; x-1

Verified step by step guidance

1. Identify the divisor polynomial and set it equal to zero to find the root. For the divisor \(x - 1\), set \(x - 1 = 0\) which gives \(x = 1\).
2. Use synthetic division to divide the first polynomial \(x^3 - 5x^2 + 3x + 1\) by \(x - 1\). Write down the coefficients of the dividend polynomial: 1 (for \(x^3\)), -5 (for \(x^2\)), 3 (for \(x\)), and 1 (constant term).
3. Set up synthetic division by placing the root \(1\) to the left and the coefficients in a row: \(1, -5, 3, 1\). Begin the synthetic division process by bringing down the first coefficient as is.
4. Multiply the root by the number just brought down and write the result under the next coefficient. Add the column and continue this process across all coefficients.
5. Examine the final number obtained after the last addition (the remainder). If the remainder is zero, then \(x - 1\) is a factor of the polynomial by the Factor Theorem; 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 means that substituting c into the polynomial yields zero, confirming that (x - c) divides the polynomial exactly without remainder.

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 and less error-prone than long division, and helps find remainders and quotient polynomials.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. Determining if one polynomial is a factor of another requires checking if division results in zero remainder, which confirms exact divisibility and helps break down complex polynomials into simpler components.