Key takeaway: Factoring turns a sum of terms into a multiplication of smaller pieces, reversing expansion so that the pieces multiply back to exactly what you began with.
Grade 9 – Algebra 1A-SSE.A.2Expressions and Equations
Definition of Factoring
Factoring breaks an expression apart into factors that multiply to give it. Just as 63 can be written as \(7 \times 9\), the trinomial \(x^{2} + 16x + 63\) can be written as \((x + 7)(x + 9)\). The factored form shows the zeros of the expression and makes equations easier to solve.
Begin with any greatest common factor. After that, scan for a known pattern, such as two squares subtracted or a squared binomial. For a trinomial with leading coefficient 1, hunt for a pair of integers whose product is the constant and whose sum is the middle coefficient. Some polynomials resist all of these tries; they are called prime over the integers. Practice with factoring builds fast number sense about products and sums.
Formula and notation
Factoring reverses multiplication:
\[x^{2} + bx + c = (x + p)(x + q) \quad\text{where } pq = c \text{ and } p + q = b\]
Always verify by multiplying the factors back together with the distributive property.
Key facts
- Check for a greatest common factor before trying any other pattern.
- For x² + bx + c, look for two integers with product c and sum b.
- A completely factored expression cannot be factored further.
- Multiplying the answer back out is the fastest way to check your factoring.
Factoring example
Factor \(x^{2} + 16x + 63\).
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Find two numbers with product 63 and sum 16. The pairs of 63 are 1 and 63, 3 and 21, 7 and 9.
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Only 7 and 9 add up to 16.
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Write the factors: \((x + 7)(x + 9)\). Check: \(x^{2} + 9x + 7x + 63 = x^{2} + 16x + 63\).
\(x^{2} + 16x + 63 = \) \((x + 7)(x + 9)\).
Common mistake
Watch out
Ida picks 3 and 21 because they multiply to 63 and writes \((x + 3)(x + 21)\).
Why it is wrong
The product matches, but the sum must also match the middle coefficient: \(3 + 21 = 24\), not 16. Multiplying out her answer gives \(x^{2} + 24x + 63\). Test both conditions, or multiply back to check.
Related terms and lessons
- Factoring Out the GCF
- Difference of Squares
- Perfect Square Trinomial
- Zero Product Property
- FOIL Method
- Trinomial
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