Unit 5 · Lesson 3 of 5

Adding, subtracting, and multiplying polynomials

About 25 minutes · 5 steps · one idea: combine like terms, and multiply every term by every term

You'll be able to: add, subtract, and multiply polynomials and write the answer in standard form.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.

Simplify \((3x^2)(4x^5)\).

Multiply the numbers: \(3 \cdot 4 = 12\). Same base, so add the exponents: \(x^2 \cdot x^5 = x^{2+5} = x^7\).

Answer: \(12x^7\).

Simplify \(5(x - 2) - 3(x + 4)\).

Distribute: \(5x - 10 - 3x - 12\). The \(-3\) multiplies both \(x\) and \(4\).

Combine like terms: \(5x - 3x = 2x\) and \(-10 - 12 = -22\).

Answer: \(2x - 22\).

Step 2 of 5The idea

A polynomial is a sum of terms like \(4x^3\), \(-5x\), or \(7\), where every exponent is a whole number. Example: \(4x^3 - 5x + 7\).

Some names: one term is a monomial, two terms is a binomial, three terms is a trinomial. The degree is the highest exponent. Standard form means the terms are written from highest exponent to lowest.

Adding and subtracting is just combining like terms. Multiplying uses the distributive property over and over, then exponent rules from the last unit.

Add. Drop the parentheses and combine like terms.

Subtract. The minus sign goes to every term in the second polynomial: \(-(a - b + c) = -a + b - c\). Then combine like terms.

Multiply. Multiply every term of the first by every term of the second, then combine like terms. For two binomials this is FOIL: First, Outer, Inner, Last.

A box (a small table) also works: put one polynomial's terms across the top, the other's down the side, and fill in each product.

Why this shows up on the placement test: "simplify" and "which is equivalent" questions often hide a subtraction of a whole polynomial or a product of two binomials.

Step 3 of 5Worked examples

Example 1: \((4x^2 - 3x + 5) - (x^2 - 6x - 2)\)

  1. Flip every sign in the second polynomial: \(4x^2 - 3x + 5 - x^2 + 6x + 2\). \(-(-6x) = +6x\) and \(-(-2) = +2\).
  2. \(x^2\) terms: \(4x^2 - x^2 = 3x^2\).
  3. \(x\) terms: \(-3x + 6x = 3x\).
  4. Numbers: \(5 + 2 = 7\). Answer: \(3x^2 + 3x + 7\).

Example 2: \((2x + 3)(x - 4)\), then \((x + 2)(x^2 - 3x + 1)\)

  1. FOIL. First: \(2x \cdot x = 2x^2\). Outer: \(2x \cdot (-4) = -8x\). Inner: \(3 \cdot x = 3x\). Last: \(3 \cdot (-4) = -12\).
  2. Combine the middle terms: \(-8x + 3x = -5x\). Answer: \(2x^2 - 5x - 12\).
  3. For the second one, multiply \(x\) by each term: \(x^3 - 3x^2 + x\). FOIL only covers two binomials; here each of 2 terms meets each of 3 terms, so 6 products.
  4. Multiply \(2\) by each term: \(2x^2 - 6x + 2\).
  5. Combine: \(x^3 + (-3x^2 + 2x^2) + (x - 6x) + 2 = x^3 - x^2 - 5x + 2\).

Step 4 of 5Practice

Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went. Use ^ for powers, like 3x^2 - 5x + 1.

What is the degree of \(5x^3 - 2x^4 + 7x - 1\)?

The degree is the highest exponent, wherever it appears. The exponents are 3, 4, 1, and 0.

Answer: \(4\).

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

Combine like terms: \(3x^2 + x^2 = 4x^2\), \(2x - 5x = -3x\), \(-7 + 4 = -3\).

Answer: \(4x^2 - 3x - 3\).

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

Flip every sign in the second polynomial: \(5x^2 - x + 3 - 2x^2 - 4x + 6\).

Combine: \(5x^2 - 2x^2 = 3x^2\), \(-x - 4x = -5x\), \(3 + 6 = 9\).

Answer: \(3x^2 - 5x + 9\).

Multiply out: \(-3x(2x^2 - 5x + 4)\).

Multiply \(-3x\) by each term: \(-3x \cdot 2x^2 = -6x^3\), \(-3x \cdot (-5x) = 15x^2\), \(-3x \cdot 4 = -12x\).

Answer: \(-6x^3 + 15x^2 - 12x\).

Multiply out: \((x + 6)(x - 2)\).

FOIL: \(x^2 - 2x + 6x - 12\).

Combine the middle terms: \(-2x + 6x = 4x\).

Answer: \(x^2 + 4x - 12\).

Multiply out: \((3x - 2)(2x + 5)\).

First: \(3x \cdot 2x = 6x^2\). Outer: \(3x \cdot 5 = 15x\). Inner: \(-2 \cdot 2x = -4x\). Last: \(-2 \cdot 5 = -10\).

Combine: \(15x - 4x = 11x\).

Answer: \(6x^2 + 11x - 10\).

Multiply out: \((x - 3)(x^2 + 2x - 5)\).

\(x\) times each term: \(x^3 + 2x^2 - 5x\).

\(-3\) times each term: \(-3x^2 - 6x + 15\).

Combine: \(x^3 + (2x^2 - 3x^2) + (-5x - 6x) + 15\).

Answer: \(x^3 - x^2 - 11x + 15\).

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

  1. \(x^2 - 10x - 1\)
  2. \(x^2 - 2x + 1\)
  3. \(x^2 - 2x - 1\)
  4. \(3x^2 - 2x - 1\)

Distribute: \(2x(x - 3) = 2x^2 - 6x\).

Flip every sign in the second polynomial: \(2x^2 - 6x - x^2 + 4x - 1\).

Combine: \(2x^2 - x^2 = x^2\), \(-6x + 4x = -2x\), and \(-1\). Answer: \(x^2 - 2x - 1\), choice (c).

(a) comes from not flipping \(-4x\); (b) from not flipping \(+1\); (d) from adding \(x^2\) instead of subtracting it.

Step 5 of 5Wrap-up

You can now add and subtract polynomials (flip every sign after a minus) and multiply them by multiplying every term by every term, then combining like terms.

Read or watch next: OpenStax Intermediate Algebra 2e, 5.1 Add and Subtract Polynomials (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Algebra 2: Adding and subtracting polynomials (short videos). Quick rules: Exponents and polynomials cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "When I multiply a binomial by a trinomial, how do I keep track of all six products?" It knows this course and can explain things another way.