Add, Subtract, and Simplify Polynomials
We add and subtract polynomials by grouping like terms. Below, we define like terms, demonstrate how to group them correctly, and show how to add and subtract them to completely simplify polynomial expressions.
An online calculator to expand and simplify polynomials may be used to verify your answers.
Why this matters: Combining like terms is a foundational skill in algebra. It is essential for solving equations, factoring polynomials, and simplifying expressions in calculus. Mastering this skill early makes more advanced topics much easier.
Like Terms in a Polynomial
Like terms in a polynomial are terms that have the exact same variable (or variables) raised to the exact same power.
Example 1: Identifying Like Terms
- \(2x\) and \(-6x\) are like terms because they both have the same variable \(x\) raised to the same power of \(1\).
- \(2x^2\) and \(3x\) are not like terms because the variable \(x\) has a power of \(2\) in the first term and a power of \(1\) in the second.
- \(-x^3\) and \(3x^3\) are like terms because the variable \(x\) has the same power of \(3\) in both terms.
- \(-yx^2\) 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 (\(x\)).
- \(-y^2x^4\) and \(3x^4y^2\) are like terms because they contain the exact same variables (\(x\) and \(y\)), each raised to the same respective powers.
Grouping Like Terms in a Polynomial
When simplifying, we factor out the common variable component and compute the numbers (coefficients).
Example 2
Add and simplify: \( 4x + 6x \)
Solution: Factor \(x\) out of both terms, then add the numbers.
\[ 4x + 6x = x(4 + 6) = 10x \]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.
\[ 3y^2x^4 - 4x^4y^2 = y^2x^4(3 - 4) = -1y^2x^4 = -y^2x^4 \]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: \( 3x^2 + 2x + 7 \) and \( 7x^2 - 9x - 5 \)
Solution: Group the like terms together, then simplify each group.
\[ (3x^2 + 7x^2) + (2x - 9x) + (7 - 5) \] \[ = 10x^2 - 7x + 2 \]Example 6
Add and simplify: \( (3x^2 + 2xy + 7x - y^2) + (-x^2 + 4xy + 10x - 2y^2) \)
Solution: Group the like terms.
\[ (3x^2 - x^2) + (2xy + 4xy) + (7x + 10x) + (-y^2 - 2y^2) \] \[ = 2x^2 + 6xy + 17x - 3y^2 \]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, distribute it by multiplying 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.
\[ (5x^4) + (2x^3 + 7x^3 + x^3) + (-8x^2 - 9x^2 - 2x^2) + (-10x - 5x + 3x) + (2 + 3 - 7) \] \[ = 5x^4 + 10x^3 - 19x^2 - 12x - 2 \]Common Mistakes to Avoid
- Forgetting to distribute the negative sign. When subtracting a polynomial in parentheses, every term inside changes sign. For example: \[ (3x + 5) - (2x - 4) = 3x + 5 - 2x + 4 = x + 9 \] Not \( 3x + 5 - 2x - 4 \).
- Combining terms with different exponents. \( x^2 \) and \( x \) are not like terms. \( 3x^2 + 2x \) cannot be simplified further.
- Combining terms with different variables. \( 2xy \) and \( 3x \) are not like terms because the first has both \( x \) and \( y \), while the second has only \( x \).
- Ignoring the order of variables. \( 3x^2y \) and \( -5yx^2 \) are like terms because multiplication is commutative (\( x^2y = yx^2 \)).
Practice Questions
Expand (if necessary), then add or subtract and simplify. Try each question on your own before opening the solution.
Question A
\( 2x - 3x + 3y - y + 4x - 5y \)
Solution to Question A
Group the \(x\) terms and the \(y\) terms:
\[ (2 - 3 + 4)x + (3 - 1 - 5)y \] \[ = 3x - 3y \]Question B
\( -2(x - 3) - 4(x + y + 2) - 5y \)
Solution to Question B
First, distribute the coefficients to remove parentheses:
\[ -2x + 6 - 4x - 4y - 8 - 5y \]Group like terms:
\[ (-2x - 4x) + (-4y - 5y) + (6 - 8) \] \[ = -6x - 9y - 2 \]Question C
\( 2x^2 - 3x - 9y^2 - y + 4x - 5y^2 - 5y \)
Solution to Question C
Group like terms:
\[ (2x^2) + (-3x + 4x) + (-9y^2 - 5y^2) + (-y - 5y) \] \[ = 2x^2 + x - 14y^2 - 6y \]Question D
\( (-2x + 4y - 2) - 3(x - 6y - 1) + 5(x - y) \)
Solution to Question D
Expand and distribute to remove parentheses:
\[ -2x + 4y - 2 - 3x + 18y + 3 + 5x - 5y \]Group like terms:
\[ (-2x - 3x + 5x) + (4y + 18y - 5y) + (-2 + 3) \] \[ = 0x + 17y + 1 \] \[ = 17y + 1 \]Question E
\( (5x^2 + 4y - 2) - 5(x - 6y^2 - 1) + 6(2x - y) \)
Solution to Question E
Expand and distribute to remove parentheses:
\[ 5x^2 + 4y - 2 - 5x + 30y^2 + 5 + 12x - 6y \]Group like terms:
\[ 5x^2 + 30y^2 + (-5x + 12x) + (4y - 6y) + (-2 + 5) \] \[ = 5x^2 + 30y^2 + 7x - 2y + 3 \]Frequently Asked Questions
What are like terms in a polynomial?
Like terms are terms that have the exact same variable(s) raised to the exact same power(s). For example, \( 3x^2 \) and \( -5x^2 \) are like terms, but \( 3x^2 \) and \( 3x \) are not.
How do you add polynomials?
To add polynomials, group like terms together and then add their coefficients. For example, \( (3x^2 + 2x) + (5x^2 - 7x) = (3 + 5)x^2 + (2 - 7)x = 8x^2 - 5x \).
How do you subtract polynomials?
To subtract polynomials, distribute the minus sign to every term in the second polynomial (i.e., change the sign of each term), then group and combine like terms.
What is the most common mistake when subtracting polynomials?
The most common mistake is forgetting to distribute the negative sign to every term in the second polynomial. For example, \( (3x + 5) - (2x - 4) = 3x + 5 - 2x + 4 = x + 9 \), not \( 3x + 5 - 2x - 4 \).
Can you combine \( x^2 \) and \( x \) terms?
No. \( x^2 \) and \( x \) are not like terms because the variable is raised to different powers. You can only combine terms with the exact same variable and exponent.