College Algebra - Week 7

Polynomials

Polynomials are the building blocks of algebra. From constants and linear expressions to cubics and beyond, this week you master the vocabulary, arithmetic operations, and multiplication techniques for polynomials. Special products — including FOIL, the square of a binomial, and the difference of squares — appear constantly in factoring, calculus, and beyond.

Learning Objectives

- Identify degree, leading coefficient, and classify polynomials by term count. - Add and subtract polynomials by combining like terms. - Multiply monomials and apply the rules of exponents. - Multiply polynomials using the distributive property and FOIL. - Apply the special product formulas: (a+b)^2, (a-b)^2, and (a+b)(a-b). - Divide a polynomial by a monomial. - Perform polynomial long division.

1. Polynomial Vocabulary

A polynomial is a sum of terms of the form ax^n where a is a real number and n is a non-negative integer. Examples: - Monomial: 5x^3 (one term) - Binomial: 3x^2 - 7 (two terms) - Trinomial: x^2 + 4x - 9 (three terms) Degree: the highest power of the variable. The degree of 4x^5 - 2x^3 + x is 5. Leading coefficient: the coefficient of the highest-degree term. Constant term: the term with no variable (degree 0).

2. Rules of Exponents

These rules must be memorized: 1. Product rule: x^a * x^b = x^(a+b) 2. Quotient rule: x^a / x^b = x^(a-b) 3. Power rule: (x^a)^b = x^(ab) 4. Power of product: (xy)^n = x^n * y^n 5. Zero exponent: x^0 = 1 (x not 0) 6. Negative exponent: x^(-n) = 1/x^n Examples: x^3 * x^5 = x^8 (x^2)^4 = x^8 (2x^3)^2 = 4x^6 6x^5 / 2x^2 = 3x^3

3. Adding and Subtracting Polynomials

Add or subtract polynomials by combining like terms. Like terms have the same variable raised to the same power. Example 1: Add (3x^2 + 5x - 2) + (x^2 - 3x + 7). Group like terms: (3x^2 + x^2) + (5x - 3x) + (-2 + 7) = 4x^2 + 2x + 5. Example 2: Subtract (4x^2 - x + 3) - (2x^2 + 5x - 1). Distribute the minus sign: 4x^2 - x + 3 - 2x^2 - 5x + 1. Combine: 2x^2 - 6x + 4. Key: distributing the minus sign means EVERY term of the second polynomial changes sign.

4. Multiplying Monomials

Multiply coefficients together and add exponents for matching variables. Example 1: (3x^2)(5x^4) = 15x^6 Example 2: (2x^3y)(4xy^2) = 8x^4y^3 Example 3: (-2ab^2)(3a^3b) = -6a^4b^3 Remember: when multiplying like bases, add the exponents.

5. Multiplying a Polynomial by a Monomial

Distribute the monomial to EVERY term in the polynomial. Example 1: 3x(2x^2 - 5x + 1) = 6x^3 - 15x^2 + 3x Example 2: -2x^2(x^3 - 4x + 7) = -2x^5 + 8x^3 - 14x^2 Example 3: 5xy(3x^2 - 2xy + y^2) = 15x^3y - 10x^2y^2 + 5xy^3

6. Multiplying Binomials: FOIL

FOIL: First, Outer, Inner, Last. (a + b)(c + d) = ac + ad + bc + bd Example 1: (x + 3)(x + 5) F: x*x = x^2 O: x*5 = 5x I: 3*x = 3x L: 3*5 = 15 Result: x^2 + 8x + 15 Example 2: (2x - 3)(x + 4) F: 2x^2, O: 8x, I: -3x, L: -12 Result: 2x^2 + 5x - 12

7. Multiplying Larger Polynomials

Distribute each term of one polynomial to every term of the other. Example: (x + 2)(x^2 - 3x + 4) Distribute x: x^3 - 3x^2 + 4x Distribute 2: 2x^2 - 6x + 8 Add: x^3 - x^2 - 2x + 8 Example 2: (2x - 1)(3x^2 + x - 5) Distribute 2x: 6x^3 + 2x^2 - 10x Distribute -1: -3x^2 - x + 5 Add: 6x^3 - x^2 - 11x + 5

8. Special Products: Squares of Binomials

These patterns appear constantly — memorize them. (a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2 Warning: (a + b)^2 is NOT a^2 + b^2! The middle term 2ab is always there. Examples: (x + 4)^2 = x^2 + 8x + 16 (2x - 3)^2 = 4x^2 - 12x + 9 (x + y)^2 = x^2 + 2xy + y^2

9. Special Products: Difference of Squares

(a + b)(a - b) = a^2 - b^2 The middle terms cancel! This is one of the most useful patterns in algebra. Examples: (x + 5)(x - 5) = x^2 - 25 (3x + 2)(3x - 2) = 9x^2 - 4 (2x + 7y)(2x - 7y) = 4x^2 - 49y^2 Recognizing this pattern in reverse is the key to factoring next week.

10. Dividing a Polynomial by a Monomial

Divide each term of the polynomial by the monomial. Example: (12x^4 - 8x^3 + 4x^2) / 4x^2 = 12x^4/(4x^2) - 8x^3/(4x^2) + 4x^2/(4x^2) = 3x^2 - 2x + 1 Example 2: (15x^3y^2 - 9xy) / 3xy = 5x^2y - 3

11. Polynomial Long Division

Divide x^2 + 5x + 6 by (x + 2). Step 1: Divide leading terms: x^2 / x = x. Write x on top. Step 2: Multiply x * (x + 2) = x^2 + 2x. Subtract: (x^2 + 5x + 6) - (x^2 + 2x) = 3x + 6. Step 3: Divide: 3x / x = 3. Write +3 on top. Step 4: Multiply 3 * (x + 2) = 3x + 6. Subtract: (3x + 6) - (3x + 6) = 0. Answer: x + 3 Verify: (x + 2)(x + 3) = x^2 + 5x + 6. Correct.

12. Common Mistakes

- (a + b)^2 ≠ a^2 + b^2. Always expand fully. - When subtracting polynomials, NOT distributing the minus to every term. - Adding exponents when they should be multiplied: x^2 * x^3 = x^5, not x^6. - In FOIL, forgetting the Outer or Inner terms. - In long division, forgetting to subtract (not add) at each step.