Lesson 6.3: Factor Special Products

New Concept

This lesson gives you a factoring toolkit! Learn to spot special patterns like perfect square trinomials and differences of squares or cubes. Recognizing these makes factoring much faster and simpler than general methods.

What’s next

Now, let's put these patterns to use. You'll dive into interactive examples and practice cards for each type of special product to build your factoring speed.

Property

If $a$ and $b$ are real numbers,

To factor a perfect square trinomial, first confirm the trinomial fits the pattern: check if the first term is a perfect square ($a^2$), the last term is a perfect square ($b^2$), and the middle term is $2ab$. If it matches, write the result as the square of a binomial, $(a+b)^2$ or $(a-b)^2$, depending on the sign of the middle term.

Examples

• To factor $16x^2 + 24x + 9$, we see it fits the pattern $(4x)^2 + 2(4x)(3) + (3)^2$. This factors to $(4x + 3)^2$.

• To factor $25y^2 - 60y + 36$, notice the negative middle term. The pattern is $(5y)^2 - 2(5y)(6) + (6)^2$, which factors to $(5y - 6)^2$.

Property

If $a$ and $b$ are real numbers,

To use this pattern, verify that the binomial is a difference and that both terms are perfect squares. Write the terms as squares, $(a)^2 - (b)^2$, and then write the product of the conjugates, $(a - b)(a + b)$. Note that a sum of squares, $a^2 + b^2$, is prime and cannot be factored.

Examples

• To factor $49x^2 - 1$, recognize it as a difference of squares: $(7x)^2 - (1)^2$. This factors into the product of conjugates: $(7x - 1)(7x + 1)$.

• To factor $100a^2 - 81b^2$, write the terms as squares: $(10a)^2 - (9b)^2$. The factors are $(10a - 9b)(10a + 9b)$.

Property

If $a$ and $b$ are real numbers,

The sign of the binomial factor matches the sign in the original binomial. The sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. The trinomial factor in this pattern cannot be factored further.

Examples

• To factor $y^3 + 27$, recognize it as a sum of cubes, $y^3 + 3^3$. Using the pattern, this factors to $(y + 3)(y^2 - 3y + 9)$.

• To factor $8m^3 - 125n^3$, write the terms as cubes: $(2m)^3 - (5n)^3$. This difference of cubes factors to $(2m - 5n)(4m^2 + 10mn + 25n^2)$.