Lesson 5: Simon's Favorite Factoring Trick

Property

For expressions of the form $ab + ay + bx + xy$, add the constant $ab$ to both sides to create a factorable form:

Examples

Property

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

HOW TO: Factor by grouping.
Step 1. Group terms with common factors.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common factor from the expression.
Step 4. Check by multiplying the factors.

Examples

• Factor $ab + 5a + 3b + 15$. Group the terms: $(ab + 5a) + (3b + 15)$. Factor the GCF from each group: $a(b+5) + 3(b+5)$. Factor out the common binomial: $(b+5)(a+3)$.
• Factor $x^2 + 2x - 5x - 10$. Group the terms: $(x^2 + 2x) + (-5x - 10)$. Factor GCFs: $x(x+2) - 5(x+2)$. Factor out the common binomial: $(x+2)(x-5)$.
• Factor $mn - 8m + 4n - 32$. Group the terms: $(mn - 8m) + (4n - 32)$. Factor GCFs: $m(n-8) + 4(n-8)$. Factor out the common binomial: $(n-8)(m+4)$.

Property

For Diophantine equations containing product terms like $xy + ax + by = c$, apply Simon's Favorite Factoring Trick by adding $ab$ to both sides: $xy + ax + by + ab = c + ab$, which factors as $(x + a)(y + b) = c + ab$.

Examples