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Factorising polynomials by grouping

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Factoring Polynomials by Grouping

Factoring polynomials by grouping is a method for four-term polynomials: group the terms in pairs, factor the greatest common factor from each pair, then factor out the binomial the pairs share. Learn the steps with a worked example and how to check your answer.

Factoring polynomials by grouping

Factoring by grouping is a method for factoring a polynomial with four (or more) terms by splitting it into pairs, factoring each pair, and then factoring out what the pairs share.

Factoring by grouping Steps to factor a four-term polynomial by grouping: group the first two terms and the last two terms, factor the greatest common factor from each group, then factor out the common binomial. x³ + 3x² + 2x + 6 1. Group:(x³ + 3x²) + (2x + 6) 2. Factor each:x²(x + 3) + 2(x + 3) 3. Common factor:(x + 3)(x² + 2) Factored form: (x + 3)(x² + 2)
Group the terms in pairs, factor each pair, then factor out the shared binomial.

The steps

For x³ + 3x² + 2x + 6: group the first two terms and the last two terms — (x³ + 3x²) + (2x + 6). Factor the greatest common factor from each group: x²(x + 3) + 2(x + 3). Both groups now share the binomial factor (x + 3), so factor it out: (x + 3)(x² + 2).

When grouping works

Grouping succeeds when, after factoring each pair, the two groups share the exact same binomial. If they don't match, try reordering the terms first — the grouping isn't always in the order the polynomial was written. This method builds on prime factorization and connects to the more general factor by grouping technique used across algebra.

Checking your answer

You can always check a factoring result by multiplying the factors back out with binomial multiplication — if you get the original polynomial, the factoring is correct.

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