Factoring Trinomials in Intermediate Algebra

Factoring Trinomials

Introduction

Factoring trinomials is a fundamental skill in algebra, especially when solving quadratic equations or simplifying expressions. This section focuses on two main types of trinomials: those with a leading coefficient of 1 and those with a leading coefficient not equal to 1. Mastery of these techniques is essential for success in Intermediate Algebra.

Factoring Trinomials of the Type $x^2 + bx + c$

Understanding the Structure

• Trinomial: An algebraic expression with three terms, typically written as $x^2 + bx + c$.

• Goal: Express the trinomial as a product of two binomials.

Factoring Process

1. If necessary, rewrite the trinomial in descending order of powers of $x$.

2. Look for a common factor in all terms and factor it out first.

3. Find two numbers, $p$ and $q$, such that:

   • $p \cdot q = c$ (the product is the constant term)

   • $p + q = b$ (the sum is the coefficient of $x$)

4. Write the factorization as $(x + p)(x + q)$.

5. Check your answer by expanding the binomials using the FOIL method.

Example

• Factor $x^2 - 8x + 15$.

  • Find $p$ and $q$ such that $p \cdot q = 15$ and $p + q = -8$.

  • $p = -5$, $q = -3$.

  • So, $x^2 - 8x + 15 = (x - 5)(x - 3)$.

Special Cases

• If the trinomial cannot be factored using integer coefficients, it is called prime.

• Always factor out the greatest common factor (GCF) before attempting to factor the trinomial further.

Practice Problems

• Factor $x^2 + 9x + 8$.

• Factor $x^2 + 10x - 24$.

Factoring Trinomials of the Type $ax^2 + bx + c$, $a \neq 1$

Method 1: Factoring with FOIL (Trial and Error)

1. Factor out the GCF if possible and write the trinomial in descending order.

2. List possible pairs of first terms whose product is $a x^2$.

3. List possible pairs of last terms whose product is $c$.

4. Test combinations to find which pair gives the correct middle term ($bx$) when expanded.

Example

• Factor $3x^2 - 2x - 5$.

  • Possible first terms: $3x$ and $x$.

  • Possible last terms: $-5$ and $1$, $5$ and $-1$.

  • Test combinations to find the correct one: $(3x + 5)(x - 1)$ expands to $3x^2 + 2x - 5$ (not correct), try $(3x - 5)(x + 1)$, etc.

Method 2: The Grouping Method

1. Factor out the GCF and write in descending order.

2. Multiply $a \cdot c$.

3. Find two numbers $p$ and $q$ such that $p \cdot q = a \cdot c$ and $p + q = b$.

4. Rewrite the middle term $bx$ as $px + qx$.

5. Factor by grouping.

Example

• Factor $6x^2 + 7x + 2$.

  • $a \cdot c = 12$; $b = 7$.

  • Find $p = 6$, $q = 1$ ($6 \cdot 1 = 6$, $6 + 1 = 7$) Correction: $a \cdot c = 12$, so $p = 6$, $q = 1$ ($6 \cdot 1 = 6$), but $2 \cdot 6 = 12$, $2 + 6 = 8$; try $3$and $4$ ($3 \cdot 4 = 12$, $3 + 4 = 7$).

  • Rewrite: $6x^2 + 3x + 4x + 2$.

  • Group: $(6x^2 + 3x) + (4x + 2) = 3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1)$.

Summary Table: Factoring Trinomials

Type	Steps	Example
$x^2 + bx + c$	Rewrite in descending order. Factor out GCF. Find $p$ and $q$ such that $p \cdot q = c$, $p + q = b$. Write as $(x + p)(x + q)$.	$x^2 + 5x + 6 = (x + 2)(x + 3)$
$ax^2 + bx + c$, $a \neq 1$	Rewrite in descending order and factor out GCF. Multiply $a \cdot c$. Find $p$ and $q$ such that $p \cdot q = a \cdot c$, $p + q = b$. Rewrite $bx$ as $px + qx$. Factor by grouping.	$6x^2 + 7x + 2 = (3x + 2)(2x + 1)$

Common Mistakes and Tips

• Always check for a greatest common factor before factoring trinomials.

• Be careful with signs when finding $p$ and $q$.

• If a trinomial cannot be factored with integers, it is prime.

• Check your factorization by multiplying the binomials back out.

Practice Problems

• Factor $x^2 - x - 20$.

• Factor $6x^3 + 60x^2 + 126x$.

• Factor $2x^2 + 11x + 5$.

• Factor $3x^3 + 7x^2 + 2x$.

• Factor $48 - 2y - y^2$.

Summary

• Factoring trinomials is a key skill for solving quadratic equations and simplifying algebraic expressions.

• Use the appropriate method based on the form of the trinomial and always check your work.