Advanced Factoring

Chapter 36

Concept Explanation:

Factoring is the process of breaking down an expression into a product of simpler factors. Advanced factoring techniques include methods beyond simple factoring out the greatest common factor (GCF), such as factoring quadratics, factoring by grouping, and using special factoring formulas like difference of squares and perfect square trinomials.

Advanced Factoring Techniques:

Factoring Trinomials: This method is used for expressions of the form $ax^2 + bx + c$ , where you look for two numbers that multiply to $ac$ and add up to $b$ .
Difference of Squares: Used when the expression is in the form $a^2 - b^2$ , which factors as $(a - b)(a + b)$ .
Perfect Square Trinomials: If the trinomial is of the form $a^2 + 2ab + b^2$ , it factors as $(a + b)^2$ .
Factoring by Grouping: Applied when there are four or more terms, grouping pairs of terms that have a common factor.

Common Mistakes:

Ignoring the GCF: Always check if there’s a common factor that can be factored out first.
Incorrect Pairing in Grouping: When factoring by grouping, students often group terms incorrectly, which leads to an incorrect final answer.
Incorrect Use of Special Formulas: Misidentifying a trinomial as a perfect square or a difference of squares is a common error.

Helpful Tips:

Write Out the Steps: For trinomials, clearly list the factors of $ac$ and test each one to see which adds up to $b$ .
Check Your Work: After factoring, multiply the factors back out to ensure you get the original expression.

Hard Questions:

Q1: Factor $6x^2 + 5x - 6$ .

Step-by-Step Solution:

Identify $a = 6$ , $b = 5$ , and $c = -6$ . Multiply $ac = 6 \times (-6) = -36$ .
Find two numbers that multiply to $-36$ and add to $5$ . These numbers are $9$ and $-4$ .
Rewrite the middle term: