7.7Unit Review
Bring together everything from Unit 7 — polynomials, operations, special products, and factoring. Mixed practice problems and a full unit assessment to confirm mastery before moving on.
Why This Matters
Polynomials and factoring are the gateway to quadratic functions, rational expressions, and Calculus. A thorough review here ensures you're ready for Unit 8 and for the polynomial-heavy sections of the SAT.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Unit 7 · Polynomials
Unit Review — Chapters 01–06
This chapter reviews all six topics from Unit 7. Work through each section to identify areas of strength and topics that need more practice before the unit assessment.
Unit 7 — What You Learned
Unit 7 — Concept Map
- Degree, leading coefficient, standard form
- Classify: monomial, binomial, trinomial
- Evaluate polynomials for given values
- Combine like terms
- Add polynomials: align and combine
- Subtract: distribute the negative, then combine
- Distributive property
- FOIL method for binomials
- Box method for larger polynomials
- (a+b)² = a²+2ab+b²
- (a−b)² = a²−2ab+b²
- (a+b)(a−b) = a²−b²
- GCF of coefficients and variables
- Factor out GCF; write as GCF × remaining polynomial
- Negative GCF; check by distributing
- x² + bx + c: find factor pairs
- ax² + bx + c: AC method and grouping
- Difference of squares; perfect square trinomials; completely factored
Vocabulary Review
Unit 7 — Vocabulary Master List
Polynomial
A sum of terms with non-negative integer exponents. E.g. 3x² + 2x − 5.
Degree
The highest exponent in a polynomial. Determines its name (linear, quadratic, cubic…).
Standard Form
Terms written in descending order of degree.
Like Terms
Terms with the same variable(s) raised to the same power(s). Only like terms can be combined.
FOIL
First, Outer, Inner, Last — a method for multiplying two binomials.
Perfect Square Trinomial
Result of squaring a binomial: a²±2ab+b². First and last terms are perfect squares.
Difference of Squares
a²−b² = (a+b)(a−b). Only works for subtraction — a²+b² is prime.
GCF
Greatest Common Factor — the largest factor shared by all terms. Always factor this out first.
Factored Form
A polynomial written as a product of factors, e.g. (x+3)(x−2).
Prime Polynomial
A polynomial that cannot be factored over the integers, e.g. x²+4.
Special Products & Factoring — Master Reference
Special Products & Factoring — Master Reference
| Pattern | Expand → | ← Factor |
|---|---|---|
| Perfect Square (sum) | (a+b)² = a²+2ab+b² | a²+2ab+b² = (a+b)² |
| Perfect Square (diff) | (a−b)² = a²−2ab+b² | a²−2ab+b² = (a−b)² |
| Difference of Squares | (a+b)(a−b) = a²−b² | a²−b² = (a+b)(a−b) |
| Trinomial (monic) | (x+p)(x+q) = x²+(p+q)x+pq | x²+(p+q)x+pq = (x+p)(x+q) |
| GCF | a(x+y) = ax+ay | ax+ay = a(x+y) |
Mixed Problem Strategy
Mixed Problem Strategy — Decision Guide
Review Worked Examples
Ch 01 — Classify and evaluate: f(x) = 2x³ − 5x + 3 at x = 2
Degree: 3 (cubic); Leading coefficient: 2; Terms: 3 (trinomial)
f(2) = 2(2)³ − 5(2) + 3
= 2(8) − 10 + 3
= 16 − 10 + 3
Ch 02 — Subtract: (4x² + 3x − 1) − (x² − 2x + 5)
Distribute the negative: 4x² + 3x − 1 − x² + 2x − 5
Combine like terms: (4x²−x²) + (3x+2x) + (−1−5)
Ch 03 — Multiply using FOIL: (2x + 3)(x − 4)
F: 2x · x = 2x²
O: 2x · (−4) = −8x
I: 3 · x = 3x
L: 3 · (−4) = −12
Combine: 2x² + (−8x + 3x) − 12
Ch 04 — Special product: (3x − 5)²
Pattern (a−b)²: a = 3x, b = 5
a² = 9x²
−2ab = −2(3x)(5) = −30x
b² = 25
Ch 05 — Factor using GCF: 6x³ − 9x²
GCF of coefficients: GCF(6, 9) = 3
GCF of variables: GCF(x³, x²) = x²
GCF = 3x²
6x³ − 9x² = 3x²(2x − 3)
Check: 3x² · 2x = 6x³; 3x² · (−3) = −9x² ✓
Ch 06 — Factor completely: 3x² − 12
GCF = 3: factor out → 3(x² − 4)
Recognize difference of squares: a=x, b=2
3(x+2)(x−2)
Verify: 3(x²−4) = 3x²−12 ✓
Mixed — Factor then verify: x² + 5x + 6
Find p, q: multiply to 6, add to 5 → 2 and 3
Factored: (x+2)(x+3)
Verify: x²+3x+2x+6 = x²+5x+6 ✓
Top Errors to Avoid
Unit 7 — Top Errors to Avoid
Ch 02 · Problem: (3x² + 2x) − (x² − 5x)
✗ 3x² + 2x − x² − 5x = 2x² − 3x (forgot to distribute the negative)
✓ 3x² + 2x − x² + 5x = 2x² + 7x
Ch 04 · Problem: (x + 4)²
✗ x² + 16 (forgot the middle term)
✓ x² + 8x + 16 (middle term = 2·x·4 = 8x)
Ch 06 · Problem: Factor: x² + 9
✗ (x + 3)² (sum of squares is NOT factorable)
✓ x² + 9 is prime over the reals
Ch 06 · Problem: Factor: x² − 9
✗ (x − 3)² (confused difference of squares with perfect square)
✓ (x + 3)(x − 3) (difference of squares)
Common Mistakes
Combining unlike terms when adding polynomials — e.g., 3x² + 2x = 5x³.
Only add like terms (same variable, same exponent). 3x² and 2x are unlike terms.
Forgetting to distribute the negative when subtracting polynomials.
Distribute the negative to every term in the second polynomial before combining like terms.
Forgetting the middle term when squaring a binomial — e.g., (x + 6)² = x² + 36.
(x + 6)² = x² + 12x + 36. The middle term 2(x)(6) = 12x is always present.
Stopping after factoring out the GCF without checking if the remaining factor can be factored further.
After the GCF, always check for a difference of squares or factorable trinomial in the remaining expression.
Guided Review Practice
Guided Practice Video: Unit 7 Review
Watch the Unit 7 review walkthrough covering polynomials and factoring, then complete the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Classify: −7x⁴ + 2x² − x + 9. State degree, leading coefficient, and number of terms.
Hint: Degree = highest exponent. Leading coefficient = coefficient of the highest-degree term. Count the terms.
Add: (5x² − 3x + 2) + (−2x² + 7x − 4)
Hint: Align like terms: x² terms, x terms, constants. Combine coefficients.
Multiply: (x + 6)(x − 6)
Hint: Recognize the conjugate pair. Apply the difference of squares shortcut.
Expand: (2x + 1)²
Hint: Apply the perfect square pattern (a+b)². Identify a = 2x and b = 1.
Factor: x² − 11x + 28
Hint: Find two numbers that multiply to 28 and add to −11. Both will be negative.
Factor completely: 4x² − 100
Hint: Factor out the GCF first. Then check what remains for a special product pattern.
Mixed Review Practice
Classify: 6x³ − 2x + 1 (degree, leading coefficient, name)
Add: (3x² + x − 5) + (x² − 4x + 2)
Subtract: (7x² − 2x + 3) − (3x² + x − 1)
Multiply: (x + 4)(x − 3)
Multiply: (2x − 1)(3x + 5)
Expand: (x + 7)²
Expand: (x − 5)²
Expand: (4x + 3)(4x − 3)
Factor: 9x + 18
Factor: x² + 10x + 21
Factor: x² − 4x − 12
Factor: x² − 49
Factor: x² + 16x + 64
Factor completely: 2x² − 50
Factor completely: 3x² + 15x + 18