Adding and Subtracting Polynomials

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Adding and subtracting polynomials comes down to combining like terms. To add, drop the parentheses and combine terms with the same variable and exponent. To subtract, change the sign of every term in the second polynomial, then add. For example, (3x^2 + 2x) - (x^2 - 5x) = 2x^2 + 7x.

Before this lesson: You should know polynomials, combining like terms, and the distributive property.

How Adding and Subtracting Polynomials Works

Think of each kind of term as a different coin. x^2-terms are quarters, x-terms are dimes, and constants are pennies. You can add quarters to quarters, but 2 quarters and 3 dimes are not 5 of anything.

Like terms have the same variables raised to the same exponents, like 4x^2 and -7x^2. Only like terms can be combined. You add or subtract their coefficients and keep the variable part the same.

Subtracting is the tricky part. The minus sign in front of the parentheses applies to every term inside, not just the first one. It works like multiplying by −1.

Three rows. Top: (5x² − 3x + 4) − (2x² + x − 6). An arrow labeled change every sign in the second polynomial points to the middle row: 5x² − 3x + 4 − 2x² − x + 6, with the three changed terms in orange and underlined. An arrow labeled combine like terms points to the answer: 3x² − 4x + 10.
Subtracting means changing every sign in the second polynomial, then combining like terms.

Rules and Steps

OperationRuleExample
AddingDrop the parentheses. Every sign stays the same.(2x + 1) + (x - 4) = 3x - 3
SubtractingChange the sign of every term in the second polynomial, then add.(2x + 1) - (x - 4) = x + 5
-(a - b + c) = -a + b - c

Steps to Add or Subtract Polynomials

  1. If you are subtracting, change the sign of every term in the second polynomial. The minus becomes a plus.
  2. Remove the parentheses.
  3. Group like terms together. Keep each sign with the term after it.
  4. Add the coefficients of each group of like terms.
  5. Write the answer in standard form, from the highest degree to the lowest.

You can also line up like terms in columns, like adding big numbers. This is the vertical method. Leave a gap or write a 0 term when a power is missing.

Worked Examples

Example 1: Add two trinomials

Add (4x^2 + 3x - 7) + (2x^2 - 5x + 1).

\begin{aligned} &4x^2 + 3x - 7 + 2x^2 - 5x + 1 \\ &= (4x^2 + 2x^2) + (3x - 5x) + (-7 + 1) \\ &= 6x^2 - 2x - 6 \end{aligned}

The answer is 6x^2 - 2x - 6.

Example 2: Add with the vertical method

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

The first polynomial has no x^2-term and the second has no x-term. Write 0 terms so the columns line up.

x^3x^2xConstant
First3x^30x^2-2x+5
Secondx^3+4x^20x-6
Sum4x^3+4x^2-2x-1

The answer is 4x^3 + 4x^2 - 2x - 1.

Example 3: Subtract two trinomials

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

Change every sign in the second polynomial. Then combine like terms.

\begin{aligned} &5x^2 - 3x + 4 - 2x^2 - x + 6 \\ &= (5x^2 - 2x^2) + (-3x - x) + (4 + 6) \\ &= 3x^2 - 4x + 10 \end{aligned}

Check with x = 1: the first polynomial is 6 and the second is −3. So the difference should be 6 - (-3) = 9. The answer gives 3 - 4 + 10 = 9. It matches.

Example 4: Subtract with two variables

Subtract (6a^2b - 2ab + b^2) - (-a^2b + 4ab - 3b^2).

\begin{aligned} &6a^2b - 2ab + b^2 + a^2b - 4ab + 3b^2 \\ &= (6a^2b + a^2b) + (-2ab - 4ab) + (b^2 + 3b^2) \\ &= 7a^2b - 6ab + 4b^2 \end{aligned}

The terms a^2b and ab are not like terms, because the exponents on a are different.

Example 5: Word problem

A bakery sells x boxes of cookies. Its revenue in dollars is R(x) = -2x^2 + 50x and its cost is C(x) = 10x + 120. Profit is revenue minus cost. Write a polynomial for the profit, and find the profit for 10 boxes.

\begin{aligned} P(x) &= (-2x^2 + 50x) - (10x + 120) \\ &= -2x^2 + 50x - 10x - 120 \\ &= -2x^2 + 40x - 120 \end{aligned}
P(10) = -2(10)^2 + 40(10) - 120 = -200 + 400 - 120 = 80

The profit for 10 boxes is $80. Check: revenue is $300 and cost is $220, and $300 − $220 = $80.

Common Mistakes

MistakeExample of the errorFix
Changing only the first sign(4x + 3) - (x + 2) = 3x + 5The minus goes to every term: 4x + 3 - x - 2 = 3x + 1.
Combining unlike terms3x^2 + 2x = 5x^3They are not like terms. 3x^2 + 2x is already simplified.
Adding the exponentsx^2 + x^2 = x^4Add the coefficients only: x^2 + x^2 = 2x^2. You add exponents only when you multiply, as in the exponent rules.
Missing a double negative(x - 5) - (2x - 3) = -x - 8-(-3) = +3, so the answer is -x - 2.
Lining up the wrong columnsPutting 4x^2 under -2xLine up by matching powers, and write 0 terms for missing ones.

Subtracting “From” and Checking Your Answer

Word order matters. “Subtract A from B” means B - A. For example, “subtract 2x + 1 from 5x - 3” means (5x - 3) - (2x + 1) = 3x - 4.

To check any answer, pick a small number like x = 1 or x = 2. Put it into the original problem and into your answer, just like evaluating an expression. If you get different results, look for a sign error.

Adding polynomials is the first step toward multiplying them. Next you will use the distributive property to multiply binomials, and you will still combine like terms at the end.

Practice Problems

  1. Simplify (2x + 5) + (3x - 8).
  2. Simplify (x^2 - 4x + 2) + (3x^2 + x - 9).
  3. Simplify (7y^3 + 2y) + (-3y^3 + y^2 - 2y).
  4. Simplify (6x - 4) - (2x + 3).
  5. Simplify (3x^2 + 5x - 1) - (x^2 - 2x + 4).
  6. Simplify (-2m^2 + m - 8) - (-5m^2 - 3m - 8).
  7. Subtract 4x^2 - x + 3 from 9x^2 + 2x - 1.
  8. Simplify (2a^2 - 3ab + b^2) + (a^2 + 5ab - 4b^2).
  9. Simplify (x^3 + 2x - 7) - (3x^3 - x^2 + 2x).
  10. A triangle has sides x^2 + 2x, 3x - 1 and x^2 - x + 5 inches long. Write its perimeter as a polynomial. Then find the perimeter when x = 3.
Show answers
  1. (2x + 3x) + (5 - 8) = 5x - 3
  2. (x^2 + 3x^2) + (-4x + x) + (2 - 9) = 4x^2 - 3x - 7
  3. (7y^3 - 3y^3) + y^2 + (2y - 2y) = 4y^3 + y^2. The y-terms cancel.
  4. 6x - 4 - 2x - 3 = 4x - 7
  5. 3x^2 + 5x - 1 - x^2 + 2x - 4 = 2x^2 + 7x - 5
  6. -2m^2 + m - 8 + 5m^2 + 3m + 8 = 3m^2 + 4m. The constants cancel.
  7. (9x^2 + 2x - 1) - (4x^2 - x + 3) = 5x^2 + 3x - 4
  8. 3a^2 + 2ab - 3b^2
  9. x^3 + 2x - 7 - 3x^3 + x^2 - 2x = -2x^3 + x^2 - 7. The x-terms cancel.
  10. Perimeter = 2x^2 + 4x + 4. At x = 3: 2(9) + 4(3) + 4 = 34 inches. Check: the sides are 15, 8 and 11 inches, and 15 + 8 + 11 = 34.

FAQ

How do you subtract polynomials?

Change the sign of every term in the polynomial being subtracted. Then remove the parentheses and combine like terms.

Can you add polynomials with different degrees?

Yes. Combine the like terms you have and bring the others down unchanged. For example, (x^3 + 2) + (5x - 1) = x^3 + 5x + 1.

What are like terms in polynomials?

Like terms have exactly the same variables with the same exponents, such as 3x^2y and -x^2y. The coefficients can be different.

Is subtracting a polynomial the same as adding its opposite?

Yes. The opposite of a polynomial has every sign switched. So A - B is the same as A + (-B).

Summary

  • Only like terms can be combined: same variables, same exponents.
  • To add, remove the parentheses and combine like terms.
  • To subtract, change the sign of every term in the second polynomial, then add.
  • The vertical method lines up like terms in columns. Use 0 terms for missing powers.
  • Check your answer by putting a small number in for the variable.

Previous lesson: Polynomials

Next lesson: Multiplying Binomials

Back to the Algebra 1 Index

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