Factoring Trinomials with Leading Coefficient 1

Factoring Trinomials Whose Leading Coefficient is 1

Introduction

Factoring is a fundamental skill in algebra that allows us to rewrite polynomials as products of simpler expressions. In this section, we focus on trinomials of the form x2 + bx + c, where the leading coefficient (the coefficient of x2) is 1. The goal is to express these trinomials as the product of two binomials.

Key Definitions

• Trinomial: A polynomial with three terms, typically written as x2 + bx + c.

• Prime Polynomial: A polynomial that cannot be factored over a given number set (such as the integers).

• FOIL Method: A technique for multiplying two binomials: First, Outside, Inside, Last.

Strategy for Factoring Trinomials

Step-by-Step Process

To factor a trinomial of the form x2 + bx + c, follow these steps:

1. Enter x as the first term of each factor. The binomial factors will start with x: (x + ___)(x + ___).

2. List pairs of factors of the constant c. Find all pairs of integers whose product is c.

3. Try various combinations of these factors. Select the pair whose sum is equal to b (the coefficient of x).

4. Check your work by multiplying the factors using the FOIL method. The product should match the original trinomial.

If no pair of factors of c adds up to b, the trinomial is prime over the set of integers.

Examples of Factoring Trinomials

Example 1: Factor x2 + 11x + 24

• List pairs of factors of 24: (1, 24), (2, 12), (3, 8), (4, 6).

• Sum of pairs: 1+24=25, 2+12=14, 3+8=11, 4+6=10.

• The pair (3, 8) adds up to 11.

• So, $x^2 + 11x + 24 = (x + 3)(x + 8)$

• Check: $(x + 3)(x + 8) = x^2 + 8x + 3x + 24 = x^2 + 11x + 24$

Example 2: Factor a2 + 5a - 24

• List pairs of factors of -24: (-1, 24), (-2, 12), (-3, 8), (-4, 6).

• Sum of pairs: -1+24=23, -2+12=10, -3+8=5, -4+6=2.

• The pair (-3, 8) adds up to 5.

• So, $a^2 + 5a - 24 = (a - 3)(a + 8)$

• Check: $(a - 3)(a + 8) = a^2 + 8a - 3a - 24 = a^2 + 5a - 24$

Objective 1: Additional Examples

• Factor x2 + 7x + 6 Factors of 6: (1, 6), (2, 3) Sums: 1+6=7, 2+3=5 $x^2 + 7x + 6 = (x + 1)(x + 6)$

• Factor x2 + 6x + 8 Factors of 8: (1, 8), (2, 4) Sums: 1+8=9, 2+4=6 $x^2 + 6x + 8 = (x + 2)(x + 4)$

• Factor x2 + bx + c (for various values of b and c) Repeat the above process for each trinomial.

Summary Table: Factoring Trinomials

Trinomial	Factors of c	Sum	Factorization
x2 + 11x + 24	(3, 8)	11	(x + 3)(x + 8)
a2 + 5a - 24	(-3, 8)	5	(a - 3)(a + 8)
x2 + 7x + 6	(1, 6)	7	(x + 1)(x + 6)
x2 + 6x + 8	(2, 4)	6	(x + 2)(x + 4)
x2 + bx + c	Varies	Varies	Varies

Checking Your Factorization

Using the FOIL Method

• Multiply the binomials to verify the original trinomial.

• Example: $(x + 3)(x + 8) = x^2 + 8x + 3x + 24 = x^2 + 11x + 24$

Special Cases

Prime Trinomials

If no pair of integer factors of c adds up to b, the trinomial is prime and cannot be factored over the integers.

Conclusion

Factoring trinomials with a leading coefficient of 1 is a systematic process involving finding two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Mastery of this skill is essential for solving quadratic equations and simplifying algebraic expressions.