Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. x³+2x²+3; x-1

Verified step by step guidance

1. Identify the polynomials involved: the first polynomial is \(x^3 + 2x^2 + 3\) and the second polynomial is \(x - 1\).
2. Recall the Factor Theorem: a polynomial \(f(x)\) has a factor \((x - c)\) if and only if \(f(c) = 0\). Here, \(c = 1\) because the factor is \(x - 1\).
3. Evaluate the first polynomial at \(x = 1\) by substituting 1 into \(x^3 + 2x^2 + 3\), which means calculating \$1^3 + 2(1)^2 + 3$.
4. Use synthetic division to divide the first polynomial by \(x - 1\): set up synthetic division with 1 as the divisor and the coefficients of the first polynomial (1, 2, 0, 3) as the dividend. Note that the coefficient of \(x\) is 0 since it is missing.
5. Perform the synthetic division steps: bring down the first coefficient, multiply by 1, add to the next coefficient, and continue until all coefficients are processed. The remainder will tell you if \(x - 1\) is a factor (remainder 0 means it is a factor).

Key Concepts

Factor Theorem

The Factor Theorem states that a polynomial f(x) has (x - c) as a factor 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 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.

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.