5.1 Polynomial Operations

Polynomial Operations

Quick Answer: A polynomial is an algebraic expression whose terms have whole-number exponents. On the Florida Algebra 1 EOC, MA.912.AR.1.3 tests your ability to add, subtract, and multiply polynomial expressions by combining like terms and applying the distributive property.

What Is a Polynomial?

A polynomial is a sum of terms of the form ax^n, where a is a real coefficient, x is a variable, and n is a non-negative integer. Expressions with negative or fractional exponents, variables in denominators, or variables under radicals are not polynomials.

The degree of a term is the sum of the exponents on its variables; the degree of the polynomial is the largest of those sums. 3x^2 - 5x + 7 has degree 2 (quadratic), while 4x^3y + 2xy has degree 4.

Adding and Subtracting Polynomials

Addition and subtraction rely on combining like terms. Two terms are like terms when they have exactly the same variable parts — same variables raised to the same exponents. Coefficients can differ; the variable portion must match.

• 7x^2 and -3x^2 are like terms (both x^2).
• 5x^2 and 5y^2 are NOT like terms (different variables).
• 4xy and 4x are NOT like terms (different variable parts).

Procedure:

1. Remove parentheses. For subtraction, distribute the negative sign to every term in the second polynomial.
2. Group like terms.
3. Add or subtract coefficients, keeping the variable part unchanged.

Worked example (addition): (4x^2 - 3x + 8) + (2x^2 + 7x - 1).

• Combine 4x^2 + 2x^2 = 6x^2, -3x + 7x = 4x, 8 - 1 = 7.
• Result: 6x^2 + 4x + 7.

Worked example (subtraction): (5x^2 - 2x + 6) - (3x^2 + 4x - 2).

• Distribute the negative: 5x^2 - 2x + 6 - 3x^2 - 4x + 2.
• Combine: 2x^2 - 6x + 8.

Exam trap: The most common error on EOC subtraction items is forgetting to flip the sign of every term in the subtracted polynomial. "Subtract x^2 + 3x - 5 from 2x^2 - x + 4" means (2x^2 - x + 4) - (x^2 + 3x - 5) — distribute the negative across all three terms.

Multiplying Polynomials

Multiplication uses the distributive property repeatedly: each term of one polynomial multiplies each term of the other. For two binomials, students often use the FOIL pattern as a memory aid.

Monomial times polynomial: Distribute the single term across every term inside the parentheses.

• 3x(2x^2 - 5x + 1) = 6x^3 - 15x^2 + 3x.

Binomial times binomial (FOIL): (x + 4)(x - 7).

• First: x * x = x^2. Outer: x * (-7) = -7x. Inner: 4 * x = 4x. Last: 4 * (-7) = -28.
• Combine: x^2 - 3x - 28.

Binomial times trinomial: Distribute each term of the binomial across all three terms.

• (x + 2)(x^2 - 3x + 5): x * (x^2 - 3x + 5) = x^3 - 3x^2 + 5x; 2 * (x^2 - 3x + 5) = 2x^2 - 6x + 10.
• Combine: x^3 - x^2 - x + 10.

Special Products to Recognize

Florida items reward students who recognize patterns instantly:

• Difference of squares: (a + b)(a - b) = a^2 - b^2. Example: (x + 6)(x - 6) = x^2 - 36.
• Square of a sum: (a + b)^2 = a^2 + 2ab + b^2. The middle coefficient is twice the product of the terms. A frequent trap answer is a^2 + b^2 (missing the middle term).
• Square of a difference: (a - b)^2 = a^2 - 2ab + b^2.

Common Exam Traps

1. Sign error in subtraction: The subtracted polynomial has all signs flipped, not just the first term.
2. Adding unlike terms: 3x^2 + 2x cannot be combined into 5x^3 or 5x^2.
3. Multiplying exponents vs adding them: When multiplying like bases, add exponents (x^3 * x^4 = x^7, not x^12). Exponents multiply only when a power is raised to a power: (x^3)^4 = x^12.
4. Distributing a negative coefficient incorrectly: -2x(3x - 5) equals -6x^2 + 10x, not -6x^2 - 10x.
5. Forgetting the middle term: (x + 5)^2 = x^2 + 10x + 25, not x^2 + 25.

Real-World Context

Florida items frequently embed polynomial operations in a context. A typical item gives two rectangles with side lengths expressed as binomials — (x + 3) by (x + 5) and (x + 1) by (x + 2) — and asks for the difference in their areas. Multiply each pair, then subtract:

• Area 1: (x + 3)(x + 5) = x^2 + 8x + 15.
• Area 2: (x + 1)(x + 2) = x^2 + 3x + 2.
• Difference: 5x + 13.

The x^2 terms cancel deliberately in well-designed EOC items — a signature feature to watch for on test day.