Factoring Polynomials and Trinomials in College Algebra

Section 1.6: Other Types of Equations Guided Notes

Factoring Polynomials by Grouping

Factoring by grouping is a method used to factor certain polynomials, especially those with four terms. The process involves grouping terms in pairs and factoring out the greatest common factor (GCF) from each pair.

• Step 1: Group the terms in pairs.

• Step 2: Factor the GCF from each pair.

• Step 3: Factor out the common binomial factor.

Example:

• Factor $6x^2 + 2x + 3x + 1$ by grouping.

• Group: $(6x^2 + 2x) + (3x + 1)$

• Factor GCF from each pair: $2x(3x + 1) + 1(3x + 1)$

• Factor out the common binomial: $(3x + 1)(2x + 1)$

Key Terms:

• Greatest Common Factor (GCF): The largest factor that divides two or more terms.

• Binomial: A polynomial with two terms.

Factoring Trinomials with a Leading Coefficient Equal to 1

Factoring trinomials of the form $x^2 + bx + c$ is a fundamental skill in algebra. When the leading coefficient (the coefficient of $x^2$) is 1, the trinomial can often be factored into two binomials.

• Step 1: Identify two numbers that multiply to $c$ and add to $b$.

• Step 2: Write the trinomial as a product of two binomials: $(x + m)(x + n)$, where $m$ and $n$ are the numbers found in Step 1.

Examples:

• Factor $x^2 + 7x + 12$

• Find two numbers that multiply to 12 and add to 7: 3 and 4.

• So, $x^2 + 7x + 12 = (x + 3)(x + 4)$

• Factor $x^2 + 5x + 6$

• Find two numbers that multiply to 6 and add to 5: 2 and 3.

• So, $x^2 + 5x + 6 = (x + 2)(x + 3)$

• Factor $x^2 + x - 12$

• Find two numbers that multiply to -12 and add to 1: 4 and -3.

• So, $x^2 + x - 12 = (x + 4)(x - 3)$

Key Terms:

• Trinomial: A polynomial with three terms.

• Leading Coefficient: The coefficient of the term with the highest degree (here, the coefficient of $x^2$).

Summary Table: Factoring Methods

Type of Polynomial	Factoring Method	Example
Four terms	Grouping	$6x^2 + 2x + 3x + 1 = (3x + 1)(2x + 1)$
Trinomial ($x^2 + bx + c$)	Find two numbers that multiply to $c$ and add to $b$	$x^2 + 7x + 12 = (x + 3)(x + 4)$

Additional info: Factoring is a foundational skill for solving polynomial equations, simplifying expressions, and is essential for later topics such as solving quadratic equations and rational expressions.