How do you add and subtract polynomials?

To add or subtract polynomials, you first remove the brackets by distributing the signs. Then, you identify the like terms (terms with the exact same variables and exponents) and add or subtract their coefficients.

What are like terms in a polynomial?

Like terms are monomials that have the exact same variables raised to the exact same powers. For example, 3x² and -5x² are like terms, and xy² is a like term with y²x. You can only combine like terms.

How do you subtract a polynomial inside parentheses?

When a minus sign precedes a polynomial in brackets, treat it as multiplying by -1. Distribute this -1 to every single term inside the brackets, effectively flipping the sign of every term before combining them with the rest of the expression.

Add and Subtract Polynomials: Questions and Step-by-Step Solutions

The Basics: Definitions and Rules

What is a Polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, formed by adding or subtracting several monomials. Examples include:

\( x^2 + 3x - 9 \)
\( -4x^5 - 8x^3 + 3x - 7 \)
\( -\dfrac{1}{3}x^3 - 2x^2 - 5x + 1 \)
\( x^2 + 2xy + y^2 \)

Rule 1: Distributing Signs to Remove Brackets

To add or subtract polynomials, you must first remove brackets using the distributive law: \( a(b+c) = ab + ac \).

Positive Sign Preceding Brackets: A plus sign (or no sign) acts like multiplying by \( +1 \). The signs inside do not change.
Example: \( +(2x - 5) = +1(2x) + 1(-5) = 2x - 5 \)
Negative Sign Preceding Brackets: A minus sign acts like multiplying by \( -1 \). Every sign inside the brackets flips.
Example: \( -(2x - 5) = -1(2x) - 1(-5) = -2x + 5 \)

Rule 2: Adding and Subtracting Like Terms

You can only combine terms that have the exact same variables raised to the exact same powers. Order does not matter (e.g., \( x^2y^2 \) is identical to \( y^2x^2 \)).

Examples:

\( 6x + 4x - 5x = (6 + 4 - 5)x = \mathbf{5x} \)
\( -x^2 + 5x^2 - 2x^2 = (-1 + 5 - 2)x^2 = \mathbf{2x^2} \)
\( xy - 2xy + 3yx = (1 - 2 + 3)xy = \mathbf{2xy} \) (Note: \( xy \) is the same as \( yx \))

Practice Questions

Part A: Combine Like Terms

Simplify the expression: \( 2x - 2x + 9x \)
View Step-by-Step Solution
- \( 2x - 2x + 9x \)
- \( = (2 - 2 + 9)x \) (Factor the variable out and group the coefficients)
- \( = 9x \) (Simplify the numbers)
Simplify the expression: \( -x^2 + 3x^2 + x^2 \)
View Step-by-Step Solution
- \( -x^2 + 3x^2 + x^2 \)
- \( = (-1 + 3 + 1)x^2 \) (Remember that \( -x^2 \) means \( -1x^2 \) and \( x^2 \) means \( 1x^2 \))
- \( = 3x^2 \)
Simplify the expression: \( -xy + \dfrac{2}{3}xy + \dfrac{1}{2}xy \)
Hint: Find a common denominator to add the fractions.
View Step-by-Step Solution
- \( \left(-1 + \dfrac{2}{3} + \dfrac{1}{2}\right)xy \) (Factor out \( xy \))
- \( = \left(-\dfrac{6}{6} + \dfrac{4}{6} + \dfrac{3}{6}\right)xy \) (Find a common denominator of 6)
- \( = \left(\dfrac{-6 + 4 + 3}{6}\right)xy \)
- \( = \dfrac{1}{6}xy \)
Simplify the expression: \( 0.2x^3 + 2x^3 - 0.5x^3 \)
View Step-by-Step Solution
- \( (0.2 + 2 - 0.5)x^3 \) (Factor out the variable)
- \( = (2.2 - 0.5)x^3 \)
- \( = 1.7x^3 \) (Or as a fraction: \( \dfrac{17}{10}x^3 \))
Simplify the expression: \( x - 0.3x - \dfrac{1}{5}x \)
View Step-by-Step Solution
- \( \left(1 - 0.3 - \dfrac{1}{5}\right)x \) (Factor out the variable)
- \( = \left(\dfrac{10}{10} - \dfrac{3}{10} - \dfrac{2}{10}\right)x \) (Convert all to fractions with a denominator of 10)
- \( = \left(\dfrac{10 - 3 - 2}{10}\right)x \)
- \( = \dfrac{5}{10}x = \dfrac{1}{2}x \) (Or as a decimal: \( 0.5x \))

Part B: Add and Subtract Polynomials

Add the polynomials: \( (2x^2 - 2x + 1) + (x + 5) \)
View Step-by-Step Solution
- \( +1(2x^2 - 2x + 1) + 1(x + 5) \) (Identify the signs preceding the brackets)
- \( = 2x^2 - 2x + 1 + x + 5 \) (Distribute the positive signs to remove brackets)
- \( = 2x^2 + (-2x + x) + (1 + 5) \) (Group like terms together)
- \( = 2x^2 - x + 6 \) (Combine the coefficients)
Subtract the polynomials: \( (-4x^3 - 2x + 1) - (-x^3 - 5x) \)
View Step-by-Step Solution
- \( +1(-4x^3 - 2x + 1) - 1(-x^3 - 5x) \) (Identify the signs preceding the brackets)
- \( = -4x^3 - 2x + 1 + x^3 + 5x \) (Distribute the -1, flipping the signs in the second bracket)
- \( = (-4x^3 + x^3) + (-2x + 5x) + 1 \) (Group like terms)
- \( = -3x^3 + 3x + 1 \) (Simplify)
Add and subtract the polynomials: \( -(2x^3 - 2x^2 + 1) + (-x^3 - 5x^2) \)
View Step-by-Step Solution
- \( -1(2x^3 - 2x^2 + 1) + 1(-x^3 - 5x^2) \)
- \( = -2x^3 + 2x^2 - 1 - x^3 - 5x^2 \) (Distribute the signs to remove brackets)
- \( = (-2x^3 - x^3) + (2x^2 - 5x^2) - 1 \) (Group like terms)
- \( = -3x^3 - 3x^2 - 1 \) (Simplify)
Simplify: \( -(-x^4y - 2x^2 - 9) - (-yx^4 - 5x^2 + 1) \)
Hint: \( x^4y \) is the exact same term as \( yx^4 \).
View Step-by-Step Solution
- \( -1(-x^4y - 2x^2 - 9) - 1(-yx^4 - 5x^2 + 1) \)
- \( = x^4y + 2x^2 + 9 + yx^4 + 5x^2 - 1 \) (Distribute the -1 to both brackets, flipping all signs)
- \( = (x^4y + yx^4) + (2x^2 + 5x^2) + (9 - 1) \) (Group like terms)
- \( = 2x^4y + 7x^2 + 8 \) (Simplify coefficients)
Simplify the expression: \( (-x^2 - 2x) - (-x^2 - 5x + 3) + (x^2 - 4) \)
View Step-by-Step Solution
- \( +1(-x^2 - 2x) - 1(-x^2 - 5x + 3) + 1(x^2 - 4) \)
- \( = -x^2 - 2x + x^2 + 5x - 3 + x^2 - 4 \) (Distribute the signs to remove brackets)
- \( = (-x^2 + x^2 + x^2) + (-2x + 5x) + (-3 - 4) \) (Group like terms)
- \( = x^2 + 3x - 7 \) (Combine coefficients)
Subtract the polynomials: \( (x^3 - 2x^2 + 3) - \left(\dfrac{1}{4}x^3 + \dfrac{1}{2}x^2 - \dfrac{1}{3}\right) \)
View Step-by-Step Solution
- \( +1(x^3 - 2x^2 + 3) - 1\left(\dfrac{1}{4}x^3 + \dfrac{1}{2}x^2 - \dfrac{1}{3}\right) \)
- \( = x^3 - 2x^2 + 3 - \dfrac{1}{4}x^3 - \dfrac{1}{2}x^2 + \dfrac{1}{3} \) (Distribute the signs to remove brackets)
- \( = \left(1 - \dfrac{1}{4}\right)x^3 + \left(-2 - \dfrac{1}{2}\right)x^2 + \left(3 + \dfrac{1}{3}\right) \) (Factor variables out to make it easier to add fractions)
- \( = \left(\dfrac{4}{4} - \dfrac{1}{4}\right)x^3 + \left(-\dfrac{4}{2} - \dfrac{1}{2}\right)x^2 + \left(\dfrac{9}{3} + \dfrac{1}{3}\right) \) (Find common denominators)
- \( = \dfrac{3}{4}x^3 - \dfrac{5}{2}x^2 + \dfrac{10}{3} \)

Add and Subtract Polynomials - Grade 9