Lesson 3: More About Factoring

New Concept

This lesson expands your factoring toolkit. You'll learn to handle more complex polynomials by finding the greatest common factor (GCF) and using strategies like guess-and-check or the box method to factor quadratic trinomials completely.

What’s next

Get ready to master these new techniques! You'll start with practice cards on dividing powers and finding the GCF, building your skills for factoring.

Property

To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.

1. If the larger exponent occurs in the numerator, put the power in the numerator.
2. If the larger exponent occurs in the denominator, put the power in the denominator.

In symbols:

1. $\frac{a^m}{a^n} = a^{m-n}$ if $n < m$
2. $\frac{a^m}{a^n} = \frac{1}{a^{n-m}}$ if $n > m$

Property

The greatest common factor (GCF) is the largest factor that divides evenly into each term of the polynomial. It consists of the largest numerical factor and the highest power of each variable that is common to all terms. The exponent on each variable of the GCF is the smallest exponent that appears on that variable among the terms of the polynomial.

Examples

• For $15x^2y^2 - 12xy + 6xy^3$, the GCF of the coefficients 15, -12, and 6 is 3. The smallest power of $x$ is $x^1$ and of $y$ is $y^1$. The GCF is $3xy$.

• To factor $4a^3b^2 + 6ab^3 - 18a^2b^4$, the GCF is $2ab^2$. Factoring this out gives $2ab^2(2a^2 + 3b - 9ab^2)$.

Property

To factor a quadratic trinomial of the form $ax^2 + bx + c$, we reverse the FOIL method. We look for two binomials whose product matches the trinomial.

• The First terms of the binomials must multiply to $ax^2$.
• The Last terms must multiply to $c$.
• The sum of the Outer and Inner products must equal the middle term, $bx$.

Examples

• To factor $2x^2 - 7x + 3$, the first terms could be $2x$ and $x$. The last terms could be $-1$ and $-3$. The combination $(2x - 1)(x - 3)$ gives a middle term of $-6x - x = -7x$, so it is correct.

Property

To factor a trinomial in two variables of the form $ax^2 + bxy + cy^2$, we use the same methods as for single-variable trinomials. The first and last terms of the trinomial are quadratic, and the middle is a cross-term.
The factored form will look like $(px + qy)(rx + sy)$.

Examples

• To factor $x^2 + 5xy + 6y^2$, we need factors of $6y^2$ that sum to $5xy$. The factors $2y$ and $3y$ work, giving $(x + 2y)(x + 3y)$.

• To factor $a^2 - ab - 12b^2$, we need factors of $-12b^2$ that sum to $-ab$. The factors $-4b$ and $3b$ work, resulting in $(a - 4b)(a + 3b)$.

Property

To factor a polynomial completely, you should always begin by checking for a greatest common factor (GCF). After factoring out the GCF, examine the remaining polynomial factor to see if it can be factored again using other techniques.

Examples

• To factor $2b^4 + 8b^2 + 6b$, first factor out the GCF, $2b$, to get $2b(b^2 + 4b + 3)$. Then, factor the trinomial to get the final answer: $2b(b + 3)(b + 1)$.

• To factor $4a^6 - 10a^5 + 6a^4$, the GCF is $2a^4$. Factoring it out gives $2a^4(2a^2 - 5a + 3)$. Factoring the trinomial gives $2a^4(2a - 3)(a - 1)$.