Add, Subtract, and Simplify Polynomials

We add and subtract polynomials by grouping like terms. We therefore first define like terms, explain how to group them, and then show how to add and subtract them in order to simplify polynomials.

Several worked examples are included, followed by questions and their solutions.

An online calculator to expand and simplify polynomials may be used to check answers to the examples and exercises presented.

Like Terms in a Polynomial

Like terms in a polynomial are terms that have the same variable or variables raised to the same power.

Example 1

  1. \(2x\) and \(-6x\) are like terms because they both have the same variable \(x\) raised to the same power \(1\).
  2. \(2x^2\) and \(3x\) are not like terms because the variable \(x\) has power \(2\) in the first term and power \(1\) in the second.
  3. \(-x^3\) and \(3x^3\) are like terms because the variable \(x\) has the same power \(3\) in both terms.
  4. \(-yx^2\) and \(3x^2\) are not like terms because the first term has two variables \(x\) and \(y\), while the second term has only one variable \(x\).
  5. \(-y^2x^4\) and \(3x^4y^2\) are like terms because they contain the same variables \(x\) and \(y\), each raised to the same respective powers.

Grouping Like Terms in a Polynomial

Example 2

Add and simplify:

\[ 4x + 6x \]

Solution

Factor \(x\) out of both terms.

\[ 4x + 6x = x(4 + 6) \]

Then add the numbers inside the parentheses.

\[ 4x + 6x = 10x \]

Example 3

Add and simplify:

\[ - x^3 + 9x^3 \]

Solution

Factor \(x^3\) out of both terms and add the coefficients.

\[ - x^3 + 9x^3 = x^3(-1 + 9) \] \[ = 8x^3 \]

Example 4

Add and simplify:

\[ 3y^2x^4 - 4x^4y^2 \]

Solution

Factor \(y^2x^4\) out of both terms and subtract the coefficients.

\[ 3y^2x^4 - 4x^4y^2 = y^2x^4(3 - 4) \] \[ = -y^2x^4 \]

Add and Subtract Polynomials

You add and subtract polynomials by grouping like terms and then adding or subtracting their coefficients.

Example 5

Add and simplify the two polynomials:

\[ 3x^2 + 2x + 7 \quad \text{and} \quad 7x^2 - 9x - 5 \]

Solution

Group like terms together.

\[ (3x^2 + 7x^2) + (2x - 9x) + (7 - 5) \]

Then add each group.

\[ = 10x^2 - 7x + 2 \]

Example 6

Add and simplify:

\[ (3x^2 + 2xy + 7x - y^2) + (-x^2 + 4xy + 10x - 2y^2) \]

Solution

Group like terms.

\[ (3x^2 - x^2) + (2xy + 4xy) + (7x + 10x) + (-y^2 - 2y^2) \]

Add each group.

\[ = 2x^2 + 6xy + 17x - 3y^2 \]

Example 7

Add, subtract, and simplify:

\[ (5x^4 + 2x^3 - 8x^2 - 10x + 2) + (7x^3 - 9x^2 - 5x + 3) - (-x^3 + 2x^2 - 3x + 7) \]

Solution

Remove parentheses. If a minus sign precedes a set of parentheses, multiply every term inside by \(-1\).

\[ 5x^4 + 2x^3 - 8x^2 - 10x + 2 + 7x^3 - 9x^2 - 5x + 3 + x^3 - 2x^2 + 3x - 7 \]

Now group like terms.

\[ (5x^4) + (2x^3 + 7x^3 + x^3) + (-8x^2 - 9x^2 - 2x^2) + (-10x - 5x + 3x) + (2 + 3 - 7) \] \[ = 5x^4 + 10x^3 - 19x^2 - 12x - 2 \]

Questions

Expand (if necessary), then add or subtract and simplify.

  1. \(2x - 3x + 3y - y + 4x - 5y\)
  2. \(-2(x - 3) - 4(x + y + 2) - 5y\)
  3. \(2x^2 - 3x - 9y^2 - y + 4x - 5y^2 - 5y\)
  4. \((-2x + 4y - 2) - 3(x - 6y - 1) + 5(x - y)\)
  5. \((5x^2 + 4y - 2) - 5(x - 6y^2 - 1) + 6(2x - y)\)

Solutions

  1. \(3x - 3y\)
  2. \(-6x - 9y - 2\)
  3. \(2x^2 + x - 14y^2 - 6y\)
  4. \(17y + 1\)
  5. \(5x^2 + 30y^2 + 7x - 2y + 3\)

More References and Links to Polynomials