Problem 124
Question
Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.
Step-by-Step Solution
Verified Answer
To factor a 4-term polynomial by grouping, firstly, group the polynomial in pairs. Then, factor out the common factor from each grouped term. Lastly, factor out the common binomial term from the expression.
1Step 1: Group the Terms
Let's consider a general 4-term polynomial: \(ax + ay + bx + by\). Begin by grouping the terms in pairs: \((ax + ay) + (bx + by)\).
2Step 2: Factor Each Group
After grouping, factor out the common factor from each group. If \(ax + ay\) is the first group and \(bx + by\) is the second group, factor out the common factor 'a' out of the first group and 'b' out of the second group. The polynomial therefore becomes \(a(x + y) + b(x + y)\).
3Step 3: Factor out Common Binomial
Now, observe the resulting expression. It has a common factor \((x + y)\) in each term. Factor this out to give the factorized polynomial: \((a + b)(x + y)\).
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