Polynomial Operations
📖 Learn
What is a Polynomial?
A polynomial is an algebraic expression consisting of variables (also called indeterminates) and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents. Examples include: 3x + 5, x² - 4x + 7, and 2x³ + x² - 5x + 1.
Key Vocabulary
| Term | Definition | Example |
|---|---|---|
| Monomial | A polynomial with one term | 5x³, -7, 2xy |
| Binomial | A polynomial with two terms | x + 3, 2x² - 5 |
| Trinomial | A polynomial with three terms | x² + 2x + 1 |
| Degree | The highest exponent in the polynomial | x³ + 2x has degree 3 |
| Leading coefficient | The coefficient of the term with highest degree | In 4x³ + 2x, it's 4 |
| Like terms | Terms with the same variable and exponent | 3x² and -5x² are like terms |
Adding and Subtracting Polynomials
To add or subtract polynomials, combine like terms by adding or subtracting their coefficients.
- Remove parentheses (distribute the negative sign if subtracting)
- Identify like terms
- Combine like terms by adding their coefficients
- Write the result in standard form (descending order of degree)
Multiplying Polynomials
To multiply polynomials, use the distributive property: multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.
FOIL Method (for Binomials)
When multiplying two binomials (a + b)(c + d), use FOIL:
- First: Multiply the first terms (a · c)
- Outer: Multiply the outer terms (a · d)
- Inner: Multiply the inner terms (b · c)
- Last: Multiply the last terms (b · d)
Result: ac + ad + bc + bd
Special Products
| Pattern | Formula | Example |
|---|---|---|
| Square of a Sum | (a + b)² = a² + 2ab + b² | (x + 3)² = x² + 6x + 9 |
| Square of a Difference | (a - b)² = a² - 2ab + b² | (x - 4)² = x² - 8x + 16 |
| Difference of Squares | (a + b)(a - b) = a² - b² | (x + 5)(x - 5) = x² - 25 |
💡 Examples
Example 1: Adding Polynomials
Problem: Add (3x² + 5x - 2) + (2x² - 3x + 7)
Solution:
Step 1: Remove parentheses
3x² + 5x - 2 + 2x² - 3x + 7
Step 2: Group like terms
(3x² + 2x²) + (5x - 3x) + (-2 + 7)
Step 3: Combine like terms
Answer: 5x² + 2x + 5
Example 2: Subtracting Polynomials
Problem: Subtract (4x³ - 2x² + x) - (x³ + 5x² - 3x + 2)
Solution:
Step 1: Distribute the negative sign
4x³ - 2x² + x - x³ - 5x² + 3x - 2
Step 2: Group like terms
(4x³ - x³) + (-2x² - 5x²) + (x + 3x) + (-2)
Step 3: Combine like terms
Answer: 3x³ - 7x² + 4x - 2
Example 3: Multiplying Binomials (FOIL)
Problem: Multiply (2x + 3)(x - 5)
Solution:
Using FOIL:
First: 2x · x = 2x²
Outer: 2x · (-5) = -10x
Inner: 3 · x = 3x
Last: 3 · (-5) = -15
Combine: 2x² - 10x + 3x - 15
Answer: 2x² - 7x - 15
Example 4: Using Special Products
Problem: Expand (3x - 4)²
Solution:
Using (a - b)² = a² - 2ab + b² where a = 3x and b = 4:
= (3x)² - 2(3x)(4) + (4)²
= 9x² - 24x + 16
Answer: 9x² - 24x + 16
Example 5: Multiplying a Binomial by a Trinomial
Problem: Multiply (x + 2)(x² - 3x + 4)
Solution:
Distribute each term of the binomial to each term of the trinomial:
= x(x² - 3x + 4) + 2(x² - 3x + 4)
= x³ - 3x² + 4x + 2x² - 6x + 8
Combine like terms:
= x³ + (-3x² + 2x²) + (4x - 6x) + 8
Answer: x³ - x² - 2x + 8
✏️ Practice
Try these problems on your own. Choose the best answer for each question.
1. What is (5x² - 3x + 2) + (2x² + 7x - 5)?
A) 7x² + 4x - 3
B) 7x² - 4x + 3
C) 3x² + 4x - 3
D) 7x² + 10x - 3
2. Subtract: (6x³ - 4x + 1) - (2x³ + 3x² - x + 5)
A) 4x³ - 3x² - 3x - 4
B) 4x³ + 3x² - 5x + 6
C) 4x³ - 3x² - 3x + 6
D) 8x³ - 3x² - 3x - 4
3. What is (x + 4)(x - 7)?
A) x² - 3x - 28
B) x² + 3x - 28
C) x² - 11x - 28
D) x² - 3x + 28
4. Expand: (2x + 5)²
A) 4x² + 25
B) 4x² + 10x + 25
C) 4x² + 20x + 25
D) 2x² + 20x + 25
5. What is (x + 6)(x - 6)?
A) x² - 36
B) x² + 36
C) x² - 12x - 36
D) x² + 12x - 36
6. Multiply: (3x - 2)(2x + 5)
A) 6x² + 11x - 10
B) 6x² - 11x - 10
C) 6x² + 19x - 10
D) 5x² + 11x - 10
7. Expand: (x - 3)²
A) x² - 9
B) x² + 6x + 9
C) x² - 6x + 9
D) x² - 6x - 9
8. What is (x + 1)(x² + 2x - 3)?
A) x³ + 3x² - x - 3
B) x³ + 2x² - 3x - 3
C) x³ + 3x² + x - 3
D) x³ + 3x² - x + 3
9. Simplify: 2(x² - 4x + 1) - 3(x² + 2x - 5)
A) -x² - 14x + 17
B) -x² - 2x + 17
C) 5x² - 14x + 17
D) -x² - 14x - 13
10. What is the coefficient of x in the product (2x - 3)(x + 4)?
A) 5
B) -5
C) 11
D) 1
Click to reveal answers
- A) 7x² + 4x - 3 — Combine like terms: (5x² + 2x²) + (-3x + 7x) + (2 - 5)
- A) 4x³ - 3x² - 3x - 4 — Distribute negative: 6x³ - 4x + 1 - 2x³ - 3x² + x - 5
- A) x² - 3x - 28 — FOIL: x² - 7x + 4x - 28 = x² - 3x - 28
- C) 4x² + 20x + 25 — Use (a + b)² = a² + 2ab + b²
- A) x² - 36 — Difference of squares: (a + b)(a - b) = a² - b²
- A) 6x² + 11x - 10 — FOIL: 6x² + 15x - 4x - 10
- C) x² - 6x + 9 — Use (a - b)² = a² - 2ab + b²
- A) x³ + 3x² - x - 3 — Distribute: x³ + 2x² - 3x + x² + 2x - 3
- A) -x² - 14x + 17 — First distribute, then combine: 2x² - 8x + 2 - 3x² - 6x + 15
- A) 5 — FOIL gives 2x² + 8x - 3x - 12, so coefficient of x is 8 - 3 = 5
✅ Check Your Understanding
Question 1: Why is it important to distribute the negative sign when subtracting polynomials?
Reveal Answer
When subtracting polynomials, the subtraction sign applies to every term in the second polynomial. Distributing the negative sign changes the sign of each term in the polynomial being subtracted. For example, -(x² + 3x - 5) becomes -x² - 3x + 5. Forgetting to distribute the negative is one of the most common errors in polynomial operations.
Question 2: Explain why (x + 3)² is NOT equal to x² + 9. What is the correct expansion?
Reveal Answer
When squaring a binomial, you are multiplying (x + 3)(x + 3), not just squaring each term separately. Using FOIL or the special product formula (a + b)² = a² + 2ab + b², we get: x² + 2(x)(3) + 9 = x² + 6x + 9. The middle term (6x) comes from the cross-multiplication of the terms, which cannot be ignored.
Question 3: How can recognizing the difference of squares pattern save time on standardized tests?
Reveal Answer
Recognizing that (a + b)(a - b) = a² - b² allows you to skip FOIL entirely. When you see a product like (x + 7)(x - 7), you can immediately write x² - 49 without any intermediate steps. This pattern appears frequently on the SAT/ACT, and quick recognition saves valuable time. The pattern works because the "outer" and "inner" products cancel out: ax - ax = 0.
Question 4: When multiplying a binomial by a trinomial, how many individual products must you calculate before combining like terms?
Reveal Answer
You must calculate 6 individual products (2 terms × 3 terms = 6 products). Each term in the binomial gets multiplied by each term in the trinomial. For (a + b)(c + d + e), the products are: ac, ad, ae, bc, bd, and be. After finding all six products, you combine any like terms to get the final answer.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review