Factoring — Algebra 1 Unit 7 practice.
This unit covers GCF factoring, factoring trinomials, difference of squares and factoring by grouping — essential concepts for Algebra 1. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers GCF factoring, factoring trinomials, difference of squares and factoring by grouping — essential concepts for Algebra 1. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 GCF Factoring
Students must be able to identify the greatest common factor of all terms in a polynomial and factor it out. This applies to both numeric coefficients and variable parts. GCF factoring is often the first step before applying any other factoring method.
Key Points
- Find the largest number that divides all coefficients evenly
- Take the lowest power of any variable that appears in every term
- After factoring out the GCF, the expression inside parentheses should have no common factors remaining
- Always check for a GCF before attempting any other factoring strategy
Factor: 12x³ + 8x² − 4x
The GCF of 12, 8, and 4 is 4, and the lowest power of x present in all terms is x¹. Factor out 4x to get 4x(3x² + 2x − 1). Check by distributing: 4x · 3x² = 12x³, 4x · 2x = 8x², 4x · (−1) = −4x, which matches the original.
2 Factoring Trinomials
Students must factor trinomials of the form ax² + bx + c into two binomials. When a = 1, find two numbers that multiply to c and add to b. When a ≠ 1, use the AC method or trial and error with factor pairs of a · c.
Key Points
- For x² + bx + c: find factors of c that add to b, then write (x + p)(x + q)
- For ax² + bx + c with a ≠ 1: multiply a · c, find factor pairs that sum to b, then split the middle term and factor by grouping
- Signs matter: if c is positive, both factors share the sign of b; if c is negative, the larger factor takes the sign of b
- Always verify by FOILing the factored form back out
Factor: 2x² + 7x + 3
Multiply a · c = 2 · 3 = 6. Find two numbers that multiply to 6 and add to 7: those are 6 and 1. Rewrite as 2x² + 6x + x + 3, then group: 2x(x + 3) + 1(x + 3). Factor out (x + 3) to get (2x + 1)(x + 3).
3 Difference Of Squares
Students must recognize and factor expressions in the form a² − b², where both terms are perfect squares separated by subtraction. The factored form is always (a + b)(a − b). A sum of squares (a² + b²) does NOT factor over the integers.
Key Points
- Pattern: a² − b² = (a + b)(a − b)
- Both terms must be perfect squares and the operation must be subtraction
- A sum of squares a² + b² is prime (cannot be factored)
- Variables factor as perfect squares when their exponents are even: x⁴ = (x²)²
Factor: 25x² − 49
Identify that 25x² = (5x)² and 49 = 7², so this fits the difference of squares pattern. Apply the formula directly: (5x + 7)(5x − 7). Check by FOILing: 25x² − 35x + 35x − 49 = 25x² − 49. ✓
4 Factoring By Grouping
Students must factor four-term polynomials by splitting them into two groups of two terms and factoring a GCF from each group. If both groups share a common binomial factor, that binomial is factored out to complete the problem. This method also underlies the AC method for trinomials.
Key Points
- Group the first two and last two terms, then factor the GCF from each group separately
- The resulting binomials inside the parentheses must be identical for grouping to work
- If the binomials don't match, try rearranging the terms before grouping
- The final answer has two factors: the shared binomial and the binomial formed by the outer GCFs
Factor: x³ + 2x² + 3x + 6
Group as (x³ + 2x²) + (3x + 6). Factor the GCF from each group: x²(x + 2) + 3(x + 2). Since both groups contain the factor (x + 2), factor it out to get (x + 2)(x² + 3).
Questions, answered.
What is Factoring?
Factoring is Unit 7 of Algebra 1, covering GCF factoring, factoring trinomials, difference of squares and factoring by grouping.
How to study for Algebra 1 Unit 7?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.