Problem 102 - College Algebra

Question

Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.

Step-by-Step Solution

Answer

Factoring by grouping for a polynomial with four terms involves splitting the polynomial into two groups, identifying common factors within each group, factoring out the common terms, and simplifying the resulting expressions. For example, a polynomial \(ax + ay + bz + bz\) can be factored to \(a(x + y) + 2bz\).

Step 1: Understand the Polynomial

The polynomial contains four terms. For example, consider a polynomial \(ax + ay + bz + bz\).

Step 2: Group the Terms

Split the polynomial into two groups. Our polynomial can be split as follows: \((ax + ay) + (bz + bz)\).

Step 3: Identify Common Factors

Next step is to identify the common factors within each group. In this case, \(a\) and \(b\) are common to the first and second groups respectively.

Step 4: Factor the Groups

Factor out the common terms from each group, leading to: \(a(x + y) + b(z + z)\).

Step 5: Simplify the Factored Polynomial

Simplify the factors. In this case, the simplified polynomial is \(a(x + y) + 2bz\).