Problem 83 - Algebra for College Students

Question

For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the first polynomial is a factor, then factor the second polynomial. See Example 7. $$w-3, w^{3}-27$$

Step-by-Step Solution

Answer

Yes, \(w - 3\) is a factor and \(w^3 - 27 = (w - 3)(w^2 + 3w + 9)\).

Step 1: Synthetic Division Setup

Given polynomials are \(w - 3\) and \(w^3 - 27\). First, set up synthetic division with the root of the divisor. Since \(w - 3 = 0\) when \(w = 3\), use 3 as the root for synthetic division.

Step 2: Perform Synthetic Division

Write the coefficients of the dividend \(w^3 - 27\) which are \(1, 0, 0, -27\). Now, perform synthetic division: \[ \begin{array}{r|rrrr} 3 & 1 & 0 & 0 & -27 \ & & 3 & 9 & 27 \ \ \text{---} & \text{---} & \text{---} & \text{---} & \text{---} \ & 1 & 3 & 9 & 0 \ \end{array} \ \] The synthetic division result is \(1, 3, 9, 0\), corresponding to the polynomial \(w^2 + 3w + 9\). The remainder is 0, so \(w-3\) is a factor.

Step 3: Factor the Second Polynomial

Since the remainder is 0, \(w - 3\) is a factor of \(w^3 - 27\). Hence, the factorization of \(w^3 - 27\) can be written as \((w - 3)(w^2 + 3w + 9)\).

Key Concepts

Synthetic Division

Synthetic division is a shorthand method of dividing polynomials, particularly useful when dividing by a linear polynomial like \(w - 3\). It's faster and simpler compared to long division. To start, you need the root of the divisor polynomial. For \(w - 3\), the root is 3 because setting \(w - 3 = 0\) results in \(w = 3\). Next, arrange the coefficients of the dividend polynomial, \(w^3 - 27\), which are \(1, 0, 0, -27\). The synthetic division process involves systematically dropping, multiplying, and adding these coefficients. The final row provides the coefficients of the quotient polynomial and the remainder. If the remainder is zero, the divisor is indeed a factor of the dividend.

Polynomial Factorization

Once synthetic division confirms that a polynomial is a factor, you can factorize the second polynomial accordingly. Factorization is the process of breaking down a polynomial into simpler polynomials (factors) that, when multiplied, give the original polynomial. In the provided example, after applying synthetic division to \(w^3 - 27\) by \(w - 3\), we found the quotient polynomial to be \(w^2 + 3w + 9\). Since the remainder was zero, we confirmed that \(w - 3\) is a factor. Therefore, we can express \(w^3 - 27\) as the product of \(w - 3\) and \(w^2 + 3w + 9\). This factorization is helpful for solving polynomial equations, simplifying expressions, and further mathematical analysis.

Remainder Theorem

The remainder theorem states that if a polynomial \(f(w)\) is divided by another polynomial \(w - c\), the remainder of this division is simply \(f(c)\). For instance, if we want to check whether \(w - 3\) is a factor of \(w^3 - 27\), we substitute \(w = 3\) into the polynomial \(w^3 - 27\). In this case, \(f(3) = 3^3 - 27\) which equals 0. Since the remainder of 0 confirms that \(f(3) = 0\), it indicates that \(w - 3\) is indeed a factor of \(w^3 - 27\). This theorem is not only a quick check for factors but also foundational in understanding polynomial behavior and is extensively used in calculus for evaluating limits and derivatives.