Common Factors・Formula ①
Factoring means transforming a polynomial into a product. It is the reverse operation of expanding.
Factoring Out Common Factors
If there is a factor common to all terms, factor it out.
Example 1
Factor 6x²＋9x.
Answer:
The common factor is 3x.
6x²＋9x ＝ 3x(2x＋3)
Factor 6x²＋9x.
Answer:
The common factor is 3x.
6x²＋9x ＝ 3x(2x＋3)
Reverse of the Multiplication Formulas (Product of a Sum and Difference・Perfect Squares)
a²－b² ＝ (a＋b)(a－b)（product of a sum and difference）
a²＋2ab＋b² ＝ (a＋b)²（perfect square）
a²－2ab＋b² ＝ (a－b)²（perfect square）
Example 2
Factor x²－16.
Answer:
x²－4² ＝ (x＋4)(x－4)
Factor x²－16.
Answer:
x²－4² ＝ (x＋4)(x－4)
Example 3
Factor x²＋6x＋9.
Answer:
x²＋2×3×x＋3² ＝ (x＋3)²
Factor x²＋6x＋9.
Answer:
x²＋2×3×x＋3² ＝ (x＋3)²
Key Points
- First, factor out the common factor
- Form a²－b² → product of a sum and difference
- Form a²±2ab＋b² → perfect-square form
Practice Problems
- Factor 4x²y－8xy².
- Factor x²－25.
Answers・Explanations
- Answer:4xy(x－2y)
Explanation: The common factor is 4xy. 4x²y－8xy²＝4xy(x－2y). Check by expanding: 4xy×x＝4x²y、4xy×(－2y)＝－8xy² ✓ - Answer:(x＋5)(x－5)
Explanation: x²－25＝x²－5². Formula for the product of a sum and difference: a²－b²＝(a＋b)(a－b). x²－25＝(x＋5)(x－5)。