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Adding, Subtracting, Multiplying and Dividing Polynomials

Combine, multiply and divide polynomials, including FOIL, special products and factorable division.

Practice this skill 22 min Polynomial operations Leer en español
Key idea

What you'll learn

A polynomial is an expression made of terms like 3x^2, -5x and 7 added together. In this lesson you will learn to:

  • add and subtract polynomials by combining like terms,
  • multiply a polynomial by a single term, and multiply two binomials (FOIL or the box method),
  • recognize the special products (a + b)^2, (a - b)^2 and (a + b)(a - b),
  • multiply a binomial by a trinomial,
  • divide a polynomial by a single term, and divide a factorable polynomial by a binomial.

Why it matters on the GED: questions ask "Which expression is equivalent to …?" with polynomials, or ask for the area of a rectangle whose sides are expressions like 2x + 5 and x - 1. These skills are also the foundation for factoring and solving quadratic equations later in the algebra section.

Words to know

  • Monomial — one term: a number, a variable, or their product (-4x^2, 7, x).
  • Binomial — two terms (x + 5, 2x - 3).
  • Trinomial — three terms (x^2 + 7x + 12).
  • Polynomial — one or more terms whose variables have whole-number exponents (3x^3 - x + 8).
  • Degree — the highest exponent in the polynomial (5x^2 - 2x + 1 has degree 2).
  • Like terms — terms with the same variable and the same exponent (4x^2 and -x^2 are like terms; 4x^2 and 4x are not).
  • Standard form — terms written from the highest exponent to the lowest (2x^2 + 3x - 5).
  • Exponent — a small raised number showing repeated multiplication (x^3 = x \cdot x \cdot x).
  • FOIL — First, Outer, Inner, Last: a checklist for multiplying two binomials.
  • Factor — write as a product (x^2 + 7x + 12 = (x + 3)(x + 4)).

All Math words in the glossary

Adding and subtracting polynomials

Adding. Drop the parentheses and combine like terms. Only terms with the same variable and same exponent combine, and the exponent does not change.

(3x^2 + 5x - 2) + (x^2 - 7x + 6) = (3x^2 + x^2) + (5x - 7x) + (-2 + 6) = 4x^2 - 2x + 4

Notice: 3x^2 + x^2 = 4x^2, not 4x^4. When adding like terms you add the coefficients only, just as 3 apples + 1 apple = 4 apples.

Algebra tiles: three x² square tiles plus one x² square tile make four x² tiles. The count (coefficient) adds to 4, but each tile is still x², so 3x² + x² = 4x², not 4x⁴.3x² + x² = 4x² (not 4x⁴)x²x²x²+x²=x²x²x²x²3 x²-tiles and 1 x²-tile4 tiles, each still x²

Subtracting. The minus sign in front of the second parentheses applies to every term inside. Change each sign in the second polynomial, then add.

(5x^2 - 3x + 4) - (2x^2 - 6x - 1) = 5x^2 - 3x + 4 - 2x^2 + 6x + 1 = 3x^2 + 3x + 5
Term typeFirstMinus second (signs flipped)Result
x^25x^2-2x^23x^2
x-3x+6x3x
constant4+15

Lining up like terms in a table or in columns makes it hard to miss one.

Multiplying polynomials

Exponent rule: when you multiply powers of the same variable, add the exponents: x^a \cdot x^b = x^{a + b}. So x \cdot x^2 = x^3 and 3x \cdot 2x^2 = 6x^3.

Monomial × polynomial. Distribute to every term:

3x(2x^2 - 4x + 5) = 6x^3 - 12x^2 + 15x
Box model for 3x(2x² − 4x + 5): one row labelled 3x and three columns 2x², −4x and +5. The cells are 6x³, −12x² (negative, red) and 15x, so the product is 6x³ − 12x² + 15x.6x³−12x²15x2x²−4x+53x3x(2x² − 4x + 5) = 6x³ − 12x² + 15xmultiply coefficients, add exponents: 3x · 2x² = 6x³

Binomial × binomial (FOIL). Multiply each term of the first by each term of the second:

(x + 5)(x - 3) = \underbrace{x \cdot x}_{\text{First}} + \underbrace{x(-3)}_{\text{Outer}} + \underbrace{5 \cdot x}_{\text{Inner}} + \underbrace{5(-3)}_{\text{Last}} = x^2 - 3x + 5x - 15 = x^2 + 2x - 15

The box method organizes the same four products:

Box method for (x + 5)(x - 3): a 2-by-2 grid with columns labeled x and +5 and rows labeled x and -3. The cells contain x squared, 5x, -3x and -15, which add up to x squared + 2x - 15.x+5x−3x²5x−3x−15(x + 5)(x − 3) = x² + 5x − 3x − 15 = x² + 2x − 15

Special products (worth memorizing, they save time):

PatternResultExample
(a + b)^2a^2 + 2ab + b^2(x + 4)^2 = x^2 + 8x + 16
(a - b)^2a^2 - 2ab + b^2(2x - 3)^2 = 4x^2 - 12x + 9
(a + b)(a - b)a^2 - b^2(x + 4)(x - 4) = x^2 - 16

Warning: (x + 4)^2 is not x^2 + 16. Squaring means (x + 4)(x + 4), which has a middle term.

Area model of the square (x + 4)²: each side is x + 4. The square splits into an x by x piece (x²), two x by 4 pieces (4x each) and a 4 by 4 piece (16). Total x² + 8x + 16, not x² + 16.x²4x4x16x4x4(x + 4)²= x² + 4x + 4x + 16= x² + 8x + 16not x² + 16:the two 4x piecesare the middle term

Binomial × trinomial. Same idea, more products. Each of the 2 terms multiplies each of the 3 terms (6 products), then combine:

(x - 2)(x^2 + 3x - 1) = x^3 + 3x^2 - x - 2x^2 - 6x + 2 = x^3 + x^2 - 7x + 2
Box method for (x − 2)(x² + 3x − 1): rows x and −2, columns x², +3x and −1. The six cells are x³, 3x², −x, −2x², −6x and +2. Combining like terms gives x³ + x² − 7x + 2.x³3x²−x−2x²−6x+2x²+3x−1x−26 products: x³ + 3x² − x − 2x² − 6x + 2= x³ + x² − 7x + 2

Dividing polynomials

Dividing by a monomial. Divide each term by it. When dividing powers, subtract the exponents: x^a \div x^b = x^{a - b}.

\frac{6x^3 - 9x^2 + 3x}{3x} = \frac{6x^3}{3x} - \frac{9x^2}{3x} + \frac{3x}{3x} = 2x^2 - 3x + 1

Note the last term: \dfrac{3x}{3x} = 1, not 0.

Dividing a factorable polynomial by a binomial. Factor the top, then cancel the common factor.

\frac{x^2 + 7x + 12}{x + 3} = \frac{(x + 3)(x + 4)}{x + 3} = x + 4
Division as a missing side: a rectangle has area x² + 7x + 12 and one side x + 3. Filling the box with x², 3x, 4x and 12 (since 3x + 4x = 7x) shows the other side is x + 4, so (x² + 7x + 12) ÷ (x + 3) = x + 4.x²4x3x12x+4x+3other side = ?knownsidex + 3area x² + 7x + 12fill the box: x², 3x, 4x, 123x + 4x = 7x ✓(x² + 7x + 12) ÷ (x + 3)= x + 4

To factor x^2 + bx + c, find two numbers that multiply to c and add to b. For x^2 + 7x + 12: 3 \times 4 = 12 and 3 + 4 = 7.

When the leading coefficient is not 1, ask: "What times the divisor gives this polynomial?" For (2x^2 + 5x - 3) \div (2x - 1), the answer must start with x (because 2x \cdot x = 2x^2) and end with +3 (because -1 \cdot 3 = -3). Try x + 3: (2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3 ✓

Always check division by multiplying back. Quotient × divisor must give the original polynomial. (The division only makes sense when the divisor is not zero — for example x \ne -3 above.)

Formula

Rules and patterns (memorize — not on the formula sheet)

x^a \cdot x^b = x^{a + b} \qquad\qquad \frac{x^a}{x^b} = x^{a - b} (a + b)(c + d) = ac + ad + bc + bd (a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad (a + b)(a - b) = a^2 - b^2
  • x is the variable; a and b in the exponent rules are whole-number exponents.
  • In the product rules, a, b, c, d stand for any terms (numbers, variables, or both).

None of these are on the GED formula sheet. The area of a rectangle A = lw and the area of a square A = s^2, which you will often combine with them, are on the sheet.

Worked example

Example 1 (easy): Add polynomials

Simplify (4x^2 - 3x + 7) + (2x^2 + 5x - 9).

  1. Group like terms: (4x^2 + 2x^2) + (-3x + 5x) + (7 - 9).
  2. Combine: 6x^2 + 2x - 2.

The figure sums it up:

Column method for (4x² − 3x + 7) + (2x² + 5x − 9): like terms are lined up in columns for x² terms, x terms and constants. 4x² + 2x² = 6x², −3x + 5x = 2x, 7 − 9 = −2. Answer: 6x² + 2x − 2.(4x² − 3x + 7) + (2x² + 5x − 9)x² termsx termsconstantsfirst4x²−3x+7second+2x²+5x−96x²+2x−2sumAnswer: 6x² + 2x − 2

Answer: 6x^2 + 2x - 2

Check: let x = 1. Original: (4 - 3 + 7) + (2 + 5 - 9) = 8 + (-2) = 6. Answer: 6 + 2 - 2 = 6 ✓

Worked example

Example 2: Subtract polynomials

Simplify (6x^2 + 2x - 5) - (3x^2 - 4x + 1).

  1. Flip every sign in the second polynomial: 6x^2 + 2x - 5 - 3x^2 + 4x - 1.
  2. x^2-terms: 6x^2 - 3x^2 = 3x^2.
  3. x-terms: 2x + 4x = 6x.
  4. Constants: -5 - 1 = -6.

The figure sums it up:

Column method for (6x² + 2x − 5) − (3x² − 4x + 1): every sign of the second polynomial is flipped to −3x², +4x, −1 (shown in red) and the columns are added. 6x² − 3x² = 3x², 2x + 4x = 6x, −5 − 1 = −6. Answer: 3x² + 6x − 6.(6x² + 2x − 5) − (3x² − 4x + 1)x² termsx termsconstantsfirst6x²+2x−5signs flipped−3x²+4x−13x²+6x−6sumAnswer: 3x² + 6x − 6

Answer: 3x^2 + 6x - 6

Check: let x = 2. Original: (24 + 4 - 5) - (12 - 8 + 1) = 23 - 5 = 18. Answer: 12 + 12 - 6 = 18 ✓

Worked example

Example 3: Multiply two binomials

Multiply (3x - 2)(2x + 5).

  1. First: 3x \cdot 2x = 6x^2.
  2. Outer: 3x \cdot 5 = 15x.
  3. Inner: -2 \cdot 2x = -4x.
  4. Last: -2 \cdot 5 = -10.
  5. Combine the middle terms: 15x - 4x = 11x.

The figure sums it up:

Box method for (3x − 2)(2x + 5): rows 3x and −2, columns 2x and +5. Cells: First 6x², Outer 15x, Inner −4x, Last −10. The middle terms combine to 11x, giving 6x² + 11x − 10.F: 6x²O: 15xI: −4xL: −102x+53x−2middle terms:15x − 4x = 11x(3x − 2)(2x + 5) = 6x² + 11x − 10

Answer: 6x^2 + 11x - 10

Check: let x = 1. Original: (1)(7) = 7. Answer: 6 + 11 - 10 = 7 ✓

Worked example

Example 4 (word problem): Area of a patio

A rectangular patio is x + 7 feet long and x + 3 feet wide. Write a polynomial for its area in standard form.

  1. Area of a rectangle: A = lw = (x + 7)(x + 3).
  2. FOIL: x^2 + 3x + 7x + 21.
  3. Combine: x^2 + 10x + 21.

The figure sums it up:

Patio area model drawn to scale for x = 5: the length x + 7 is split into x and 7, the width x + 3 into x and 3. The four pieces are x², 7x, 3x and 21, so the area is x² + 10x + 21 square feet (96 square feet when x = 5).x²7x3x21x7x3length x + 7widthx + 3A = (x + 7)(x + 3) = x² + 3x + 7x + 21 = x² + 10x + 21Drawn for x = 5: 12 ft by 8 ft = 96 ft²

Answer: x^2 + 10x + 21 square feet

Check: if x = 5, the patio is 12 ft by 8 ft, so the area is 96 ft². The polynomial gives 25 + 50 + 21 = 96 ✓

Worked example

Example 5 (GED-hard): Find the missing side

A rectangle has area 2x^2 + 7x + 3 square meters and width 2x + 1 meters. Find an expression for its length.

  1. Length = area ÷ width: \dfrac{2x^2 + 7x + 3}{2x + 1}.
  2. The length must start with x (since 2x \cdot x = 2x^2) and its constant must be 3 (since 1 \cdot 3 = 3). Try x + 3.
  3. Multiply to test: (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3 ✓
  4. So \dfrac{(2x + 1)(x + 3)}{2x + 1} = x + 3.

The figure sums it up:

Finding a missing side: a rectangle with width 2x + 1 and area 2x² + 7x + 3. A box with rows 2x and +1 and columns x and +3 holds 2x², 6x, x and 3; since 6x + x = 7x, the length is x + 3.2x²6xx3x+32x+1length = ? → x + 3width 2x + 1area 2x² + 7x + 32x · x = 2x²1 · 3 = 36x + x = 7x ✓length = (2x² + 7x + 3) ÷ (2x + 1) = x + 3

Answer: x + 3 meters

Check: if x = 2, the area is 8 + 14 + 3 = 25 m² and the width is 5 m, so the length is 25 \div 5 = 5 m. The expression gives 2 + 3 = 5 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

The calculator cannot multiply polynomials with letters, but it can check your work, which is just as valuable.

The test-number check: choose x = 2 or x = 3 and evaluate both the original and your answer. They must give the same number.

To check (3x - 2)(2x + 5) = 6x^2 + 11x - 10 with x = 3:

  • ( 3 × 3 − 2 ) ( 2 × 3 + 5 ) enter → 77
  • 6 × 3 x² + 1 1 × 3 − 1 0 enter → 77 ✓

Use the x² key to square and the ^ key for higher powers (3 ^ 3 for 3^3). Be careful with a negative test number: type ( (−) 2 ) x² so the calculator squares -2 to get 4.

When not to use it: do the algebra by hand first. Use the calculator only to confirm, or to test answer choices when you are stuck.

Common mistake

Common mistakes

  • Adding exponents when adding terms. Wrong: 3x^2 + x^2 = 4x^4. Right: 4x^2 — exponents only add when you multiply.
  • Subtracting only the first term. Wrong: (5x^2 - 3x + 4) - (2x^2 - 6x - 1) = 3x^2 - 9x + 3. Right: flip all signs: 3x^2 + 3x + 5.
  • Squaring each term separately. Wrong: (2x - 3)^2 = 4x^2 + 9. Right: (2x - 3)(2x - 3) = 4x^2 - 12x + 9.
  • Forgetting the middle products. Wrong: (2x + 5)(x - 1) = 2x^2 - 5. Right: all four products: 2x^2 - 2x + 5x - 5 = 2x^2 + 3x - 5.
  • Sign of the last term. Wrong: (-3)(-3) = -9. Right: +9.
  • A term divided by itself becomes 0. Wrong: \dfrac{3x}{3x} = 0. Right: it equals 1.
Test strategy

GED test strategy

  • Check the first and last terms first. In a product, the first term is (first × first) and the last term is (last × last). For (2x - 3)^2: first 4x^2, last +9. That alone eliminates most wrong choices.
  • Plug in a test number. Pick x = 2, evaluate the original and each choice, and keep the one that matches. Avoid x = 0 and x = 1, which can make different choices look equal.
  • Check division by multiplying. If a choice times the divisor gives the original polynomial, it is correct.
  • Count the terms. Binomial × binomial gives up to 4 terms before combining; binomial × trinomial gives 6. If you have fewer, you skipped a product.
  • Area questions: "area" means multiply; "perimeter" means add all sides. Read which one is asked.
  • Budget your time: these questions usually take 1–2 minutes. Write the work neatly on the erasable note board; most errors are copying errors.

Check yourself

Polynomial operations

Which expression is equivalent to (3x^2 + 5x - 2) + (x^2 - 7x + 6)?

Answer choices
Hint

Combine terms with the same variable and the same exponent. Remember that x^2 means 1x^2.

Check yourself

Polynomial operations

Which expression is equivalent to (5x^2 - 3x + 4) - (2x^2 - 6x - 1)?

Answer choices
Hint

The minus sign in front of the second parentheses changes the sign of every term inside.

Summary

  • Add: combine like terms (same variable, same exponent); the exponents do not change.
  • Subtract: flip the sign of every term in the second polynomial, then add.
  • Multiply: each term times each term; add exponents (x \cdot x^2 = x^3); combine like terms.
  • Know the special products: (a \pm b)^2 = a^2 \pm 2ab + b^2 and (a + b)(a - b) = a^2 - b^2.
  • Divide: by a monomial, divide every term; by a binomial, factor and cancel.
  • Check any answer by multiplying back or by plugging in a test number.
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