Factoring UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Factoring

Identify the structure before choosing a factoring method—and always check whether the result factors further.

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Factoring

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Factoring reverses multiplication

Factoring rewrites a polynomial as a product. Start by checking for a greatest common factor. Then identify a familiar structure: a trinomial, difference of squares, perfect-square trinomial, or grouping pattern.

GCF
Remove the greatest factor shared by every term.

a² − b²
(a − b)(a + b)

x² + bx + c
Find numbers with product c and sum b.

Grouping
Create a repeated binomial factor.

GCF comes first

A factored expression is not complete if every term inside still shares a factor.

Multiply to check

Expanding your proposed factors should reproduce every term and sign in the original polynomial.

DO IT FAST

GCF → count terms → identify the pattern

Example: Factor 5x³ − 45x.

5x³ − 45x = 5x(x² − 9) = 5x(x − 3)(x + 3)

The first step removes the GCF. The remaining binomial is a difference of squares, so factoring continues.

Why it works

Using the structure in stages prevents incomplete answers and reduces guesswork.

WORKED EXAMPLES

Five forms you should recognize

1. Greatest common factor

12x³ − 18x² = 6x²(2x − 3)

Both coefficients share 6, and both terms contain at least x².

2. Trinomial

x² + 9x + 20

Find two numbers whose product is 20 and sum is 9: 4 and 5.

(x + 4)(x + 5)

3. Difference of squares

x² − 49 = x² − 7² = (x − 7)(x + 7)

This rule applies to subtraction, not x² + 49.

4. Grouping

x³ + 3x² + 2x + 6 = x²(x + 3) + 2(x + 3) = (x + 3)(x² + 2)

The goal is to create the same binomial in both groups.

5. Use factors to solve

(x − 6)(x + 2) = 0

A product equals zero when at least one factor equals zero:

x = 6 or x = −2

COMMON TRAPS

Check before you commit

• Skipping the greatest common factor
• Stopping before factoring completely
• Using difference of squares on a sum
• Choosing numbers with the correct product but wrong sum
• Losing a negative sign in the middle term
• Reading zeros without reversing the signs in the factors

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

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Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

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A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Factoring FAQ

How do I know which method to use?

Check GCF first, then count terms and inspect the pattern. Two terms may form a difference of squares; three may form a trinomial; four may factor by grouping.

How can I check my answer?

Multiply the factors. The expanded result must match every term of the original polynomial.

Is every polynomial factorable over integers?

No. Some quadratics have no integer factorization, even though they may factor using irrational or complex roots.

RELATED COMPETENCIES

Continue your mathematics review.

PolynomialsQuadratic EquationsPolynomial Roots and Discriminant

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