6.3 Study Notes

Factoring Polynomials

Introduction

Factoring polynomials is a fundamental skill in College Algebra, allowing students to simplify expressions, solve equations, and understand polynomial structure. This section focuses on factoring trinomials of the form $ax^2 + bx + c$, using systematic methods and recognizing special cases.

Factoring Trinomials of the Form $ax^2 + bx + c$

Understanding the FOIL Method

The FOIL method is used to multiply two binomials and is essential for understanding how to factor trinomials. FOIL stands for First, Outside, Inside, Last, referring to the terms multiplied in the process.

• First: Multiply the first terms in each binomial.

• Outside: Multiply the outer terms.

• Inside: Multiply the inner terms.

• Last: Multiply the last terms in each binomial.

Example:

Multiply $(3x + 2)(x + 4)$ using FOIL:

• First: $3x imes x = 3x^2$

• Outside: $3x imes 4 = 12x$

• Inside: $2 imes x = 2x$

• Last: $2 imes 4 = 8$

Sum: $3x^2 + 12x + 2x + 8 = 3x^2 + 14x + 8$

Factoring $ax^2 + bx + c$

To factor a trinomial $ax^2 + bx + c$ into $(Dx + E)(Fx + G)$:

• a is the product of the coefficients of the first terms in each binomial.

• c is the product of the coefficients of the last terms in each binomial.

• b is the sum of the products of the outside and inside coefficients.

It is important to note that b is the sum of two products, not just two numbers.

Systematic Factoring Approach

To factor $ax^2 + bx + c$:

1. List all possible pairs of factors for $a$ and $c$.

2. Test each combination to see if the sum of the outside and inside products equals $b$.

3. If no combination works, the trinomial may be prime (not factorable over the integers).

Example 1: Factoring $21x^2 - 41x + 10$

Possible factors of $21x^2$: $x imes 21x$$,$ Possible factors of $10$(since the middle term is negative, both factors should be negative):$-1 imes -10$,$-2 imes -5$

Try combinations to find the correct factorization.

Factors of 10	Resulting Binomials	Product of Outside Terms	Product of Inside Terms	Sum of Products
1, 10	$(3x - 1)(7x - 10)$	$-30x$	$-7x$	$-37x$
1, 10	$(3x - 10)(7x - 1)$	$-3x$	$-70x$	$-73x$
2, 5	$(3x - 2)(7x - 5)$	$-15x$	$-14x$	$-29x$
2, 5	$(3x - 5)(7x - 2)$	$-6x$	$-35x$	$-41x$

The correct factorization is $(3x - 5)(7x - 2)$.

Check using FOIL:

• First: $3x imes 7x = 21x^2$

• Outside: $3x imes -2 = -6x$

• Inside: $-5 imes 7x = -35x$

• Last: $-5 imes -2 = 10$

Sum: $21x^2 - 6x - 35x + 10 = 21x^2 - 41x + 10$

Example 2: Prime Trinomials

Some trinomials cannot be factored over the integers. For example, $3x^2 - 7x + 6$:

• Possible factors for $3$: $1 imes 3$

• Possible factors for $6$(negative middle term):$-1 imes -6$,$-2 imes -3$

Factors of 6	Resulting Binomials	Product of Outside Terms	Product of Inside Terms	Sum of Products
-1, -6	$(3x - 1)(x - 6)$	$-18x$	$-x$	$-19x$
-2, -3	$(3x - 2)(x - 3)$	$-9x$	$-2x$	$-11x$

No combination yields $-7x$ as the middle term, so $3x^2 - 7x + 6$ is a prime polynomial.

Factoring Out the Greatest Common Factor (GCF)

Introduction

Before factoring trinomials, always check for a greatest common factor (GCF) among all terms. Factoring out the GCF simplifies the polynomial and may make further factoring possible.

• Definition: The GCF of a set of terms is the largest expression that divides each term.

• Example: $3x^3 + 17x^2 + 10x$ GCF is $x$; factor out $x$ to get $x(3x^2 + 17x + 10)$.

Continue factoring the remaining trinomial if possible.

Factoring Perfect Square Trinomials

Definition and Recognition

A perfect square trinomial is a trinomial that results from squaring a binomial. These have a specific form and can be factored using special formulas.

• General forms:

$a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$

Examples

• Example 1: $25x^2 + 20x + 4$ - Factors of $25x^2$: $5x imes 5x$ - Factors of $4$: $2 imes 2$ - Middle term: $20x = 2 imes 5x imes 2$ - Factored form: $(5x + 2)^2$

• Example 2: $x^2 + 12x + 36$ - $x^2 + 2 imes x imes 6 + 6^2$ - Factored form: $(x + 6)^2$

• Example 3: $16x^2 - 8xy + y^2$ - $16x^2 = (4x)^2$, $y^2 = y^2$, middle term $-8xy = -2 imes 4x imes y$ - Factored form: $(4x - y)^2$

Summary Table: Factoring Trinomials

Type of Trinomial	Factoring Method	Example	Factored Form
General $ax^2 + bx + c$	Systematic trial of factor pairs	$21x^2 - 41x + 10$	$(3x - 5)(7x - 2)$
Perfect Square Trinomial	Special product formula	$x^2 + 12x + 36$	$(x + 6)^2$
Prime Trinomial	Not factorable over integers	$3x^2 - 7x + 6$	Prime

Key Points

• Always check for a GCF before factoring trinomials.

• Use the FOIL method to verify factorizations.

• Perfect square trinomials have a recognizable structure and can be factored quickly.

• If no factor pair works, the trinomial is prime.

Additional info: These notes expand on the brief points and tables in the original slides, providing full academic context, definitions, and step-by-step examples for College Algebra students.