Add and Subtract Polynomials - Grade 9

Grade 9 examples on addition and subtraction of polynomials are presented along with their detailed solutions. More questions and their solutions and detailed explanations are included.



Polynomials Examples

A polynomial is the sum of several monomials.
Example 1
These are examples of polynomials
\( \quad x^2 + 3x -9 , \quad -4x^5 - 8 x^3 + 3x - 7 , \quad -\dfrac{1}{3} x^3 - 2 x^2 - 5 x + 1 , \quad x^2 + 2xy + y^2\)

To add and subtract polynomials, you need to know how
1) to remove brackets of polynomials using the distributive law: \( \quad a(b+c) = ab + ac \quad \), which is one of the basic rules of algebra.
2) and how to add like terms.
Both techniques are explained below.



Distribute Signs Preceding Brackets in Polynomials to Remove Brackets

In what follows, we use brackets to indicate multiplication. \( \) \( \) \( \) \( \)
For example \( x \times y \quad \) may be written as \( \quad (x)(y) \quad \) or \( \quad x(y) \quad \)
1) Polynomial within brackets preceded by no sign or the plus sign such as \((2 x - 5)\) or   \( +(2 x - 5) \quad \) are the same as \( +1(2x - 5) \)

Use the distributive law: \( a(b+c) = ab + ac \quad \) to expand and hence remove brackets as follows
\( \quad \quad (2 x - 5) = \color{red}{+1}(2x - 5) = \color{red}{+1}(2x) \color{red}{+1}(- 5) = (1)(2)x +(1)(-5) = 2 x - 5 \)

\( \quad \quad +(2 x - 5) = \color{red}{+1}(2x - 5) = \color{red}{+1}(2x) \color{red}{+1}(- 5) = (1)(2)x +(1)(-5) = 2 x - 5 \)


2) Polynomial within brackets preceded by the minus sign such as \( - (2 x - 5) \quad \) is the same as \( -1(2x - 5) \)

Use the distributive law: \( a(b+c) = ab + ac \quad \) to expand and hence remove brackets as follows
\(\quad \quad - (2 x - 5) = \color{red}{-1}(2x - 5) = \color{red}-1(2x) \color{red}-1(- 5) = (-1)(2)x +(-1)(-5) = - 2 x + 5 \)



Add and Subtract Like Terms with Examples

Examples of monomials with like terms
\( - x^2 , - 6 x^2 , - x^2 \quad \) are all monomials with like terms \( x^2 \) and may be added
\( -2 y^2 x^2 , y^2 x^2 , - 2 x^2 y^2 \quad \) are all monomials with like terms \( x y^2 \) and may be added.

NOTE that the terms \( x^2 y^2 \) and \( y^2 x^2 \) in the above example are the same

Example 2
Add/subtract the like terms
a) \( 6x + 4x -5x \quad \) b) \( -x^2 + 5x^2 - 2x^2 \quad \) c) \( xy - 2xy+3yx \)

Solution to Example 2
a)
\( \begin{split} 6x + 4x -5x & = \color{red}{6}x + \color{red}{4}x \color{red}{- 5}x \quad \style{font-family:Arial; font-size: 100%}{\text{identify the coefficients}} \\\\ & = \color{red}{(6 + 4 - 5)} x \quad \style{font-family:Arial; font-size: 100%}{\text{factor the variable out and put the coefficients inside brackets}} \\\\ & = \color{red}{5} x \quad \style{font-family:Arial; font-size: 100%}{\text{add/subtract the coefficients}} \\\\ \end{split} \)
b)
\( \begin{split} -x^2 + 5x^2 - 2x^2 &= \color{red}{-1}x^2 + \color{red}{5}x^2 \color{red}{-2}x^2 \quad \style{font-family:Arial; font-size: 100%}{\text{identify the coefficients}} \\\\ & = \color{red}{(-1 + 5 - 2)} x^2 \quad \style{font-family:Arial; font-size: 100%}{\text{factor the variable out and put the coefficients inside brackets}} \\\\ & = \color{red}{2} x^2 \quad \style{font-family:Arial; font-size: 100%}{\text{add/subtract the coefficients}} \\\\ \end{split} \)
c)
\( \begin{split} xy - 2xy+3xy &= \color{red}{1}x y \color{red}{-2}y x \color{red}{+3}yx \quad \style{font-family:Arial; font-size: 100%}{\text{identify the coefficients (NOTE: x y = y x) }} \\\\ & = \color{red}{(1 - 2 + 3)} x y \quad \style{font-family:Arial; font-size: 100%}{\text{identify the coefficients, factor the variables out and put the coefficients inside brackets}} \\\\ & = \color{red}{2} xy \quad \style{font-family:Arial; font-size: 100%}{\text{add/subtract the coefficients}} \\\\ \end{split} \)

Add and Subtract Polynomials with Examples

To add and/or subtract polynomials, we add the monomials with like terms included in the polynomials to add and/or subtract.
Example 3
Add and/or subtract the following polynomials
a) \((2 x^2 + 4 x) + (4x^2 + 3x + 2) \quad \)
b) \( (3 x^3 - x^2 - 4) - ( 4 x^3 + x^2 - 5) \quad \)
c) \( - (6 x^2 y - 5 x y) + ( - 5 x y + y x^2) \)
d) \( (x^2 + 2x - 5 ) - ( -3x^2 + \dfrac{2}{3} x - 3) \)

Solution to Example 3
a)
\( \begin{split} (2 x^2 + 4 x) + (4x^2 + 3x + 2) & = \color{red}{+1} \color{green}{( 2 x^2 + 4 x )} \color{red}{+1} \color{blue}{(4x^2 + 3x + 2)} \quad \style{font-family:Arial; font-size: 100%}{\text{identify signs preceding brackets }}\\\\ & = \color{red}{+1}\color{green}{(2 x^2)} \color{red}{+1}\color{green}{(4 x)} \color{red}{+1}\color{blue}{(4 x^2)} \color{red}{+1}\color{blue}{(3 x)} \color{red}{+1}\color{blue}{(2)} \quad \style{font-family:Arial; font-size: 100%}{\text{distribute + 1 and remove brackets }}\\\\ & = \color{green}{ 2 x^2 + 4 x } + \color{blue}{4x^2 + 3x + 2} \quad \style{font-family:Arial; font-size: 100%}{\text{Multiply and simplify }}\\\\ & = (\color{green}{2x^2} + \color{blue}{4x^2}) + (\color{green}{4x} + \color{blue}{3x}) + \color{blue}{2} \quad \style{font-family:Arial; font-size: 100%}{\text{group like terms within brackets}}\\\\ & = 6x^2 + 7x + 2 \quad \style{font-family:Arial; font-size: 100%}{\text{add like terms within brackets and simplify}} \\\\ \end{split} \)

b)
\( \begin{split} (3 x^3 - x^2 - 4) - (4 x^3 + x^2 - 5) & = \color{red}{+1} \color{green}{( 3 x^3 - x^2 - 4)} \color{red}{-1} \color{blue}{(4 x^3 + x^2 - 5)} \quad \style{font-family:Arial; font-size: 100%}{\text{identify signs preceding brackets }}\\\\ & = \color{red}{+1}\color{green}{(3 x^3)} \color{red}{+1}\color{green}{(-x^2)} \color{red}{+1}\color{green}{(-4)} \color{red}{-1}\color{blue}{(4x^3)} \color{red}{-1}\color{blue}{(x^2)} \color{red}{-1}\color{blue}{(-5)} \quad \style{font-family:Arial; font-size: 100%}{\text{distribute +1 and - 1 and remove brackets .}}\\\\ & = \color{green}{ 3 x^3 - x^2 - 4} \color{blue}{-4 x^3 - x^2 + 5} \quad \style{font-family:Arial; font-size: 100%}{\text{Multiply and simplify.}}\\\\ & = (\color{green}{3x^3} \color{blue}{- 4x^3}) + (\color{green}{-x^2} \color{blue}{- x^2}) + (\color{green}{-4} \color{blue}{+ 5}) \quad \style{font-family:Arial; font-size: 100%}{\text{group like terms within brackets}} \\\\ & = -x^3 - 2x^2 + 1 \quad \style{font-family:Arial; font-size: 100%}{\text{add/subtract like terms within brackets and simplify}} \\\\ \end{split} \)

c)
\( \begin{split} - (6 x^2 y - 5 x y) + ( - 5 x y + y x^2) & = \color{red}{-1} \color{green}{( 6 x^2 y - 5 x y)} \color{red}{+1} \color{blue}{(- 5 x y + y x^2)} \quad \style{font-family:Arial; font-size: 100%}{\text{identify signs preceding brackets }}\\\\ & = \color{green}{ - 6 x^2 y + 5 x y} \color{blue}{- 5 x y + y x^2} \quad \style{font-family:Arial; font-size: 100%}{\text{distribute -1 and + 1, remove brackets and simplify.}}\\\\ & = (\color{green}{- 6 x^2 y} \color{blue}{+ y x^2}) + (\color{green}{5xy} \color{blue}{-5xy}) \quad \style{font-family:Arial; font-size: 100%}{\text{group like terms within brackets}} \\\\ & = - 5 x^2 y \quad \style{font-family:Arial; font-size: 100%}{\text{add/subtract like terms within brackets and simplify}} \\\\ \end{split} \)

d)
\( \begin{split} (x^2 + 2x - 5) - (-3x^2 + \dfrac{2}{3} x - 3) & = \color{red}{+1} \color{green}{( x^2 + 2x - 5)} \color{red}{-1} \color{blue}{(-3x^2 + \dfrac{2}{3} x - 3)} \quad \style{font-family:Arial; font-size: 100%}{\text{identify signs preceding brackets }}\\\\ & = \color{green}{ x^2 + 2x - 5} \color{blue}{+3x^2 - \dfrac{2}{3} x + 3} \quad \style{font-family:Arial; font-size: 100%}{\text{ distribute +1 and - 1, remove brackets and simplify.}}\\\\ & = (\color{green}{ x^2 } \color{blue}{+3 x^2}) + (\color{green}{2x} \color{blue}{-\dfrac{2}{3} x}) + (- \color{green}{5} \color{blue}{+3}) \quad \style{font-family:Arial; font-size: 100%}{\text{group like terms within brackets}} \\\\ & = ( \color{green}{1} \color{blue}{+ 3} ) x^2 + (\color{green}{2}\color{blue}{-\dfrac{2}{3} }) x + (\color{green}{-5} + \color{blue}{3}) \quad \style{font-family:Arial; font-size: 100%}{\text{Factor variables out to make it easier to add/subtract terms with fractions. }}\\\\ & = 4 x^2 + \dfrac{4}{3} x - 2 \quad \style{font-family:Arial; font-size: 100%}{\text{add/subtract terms within brackets and simplify}} \\\\ \end{split} \)

Questions

The solutions and detailed explanations to the questions below are included.

  1. Add and Subtract the like terms.

    1. ) \( 2x - 2x + 9x \)

    2. ) \( -x^2 + 3x^2 + x^2 \)

    3. ) \( -x y + \dfrac{2}{3} x y + \dfrac{1}{2}x y\)

    4. ) \( 0.2 x^3 + 2 x^3 - 0.5 x^3 \)

    5. ) \( x -0.3 x - \dfrac{1}{5}x \)

  2. Add and Subtract the following polynomials.

    1. ) \( (2x^2 - 2x + 1) + (x + 5) \)

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

    3. ) \( - (2x^3 - 2x^2 + 1) + ( - x^3 - 5 x^2) \)

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

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

    6. ) \( ( x^3 - 2x^2 + 3) - ( \dfrac{1}{4}x^3 + \dfrac{1}{2} x^2 - \dfrac{1}{3}) \)

Solutions and detailed explanations to the above questions are included.

More References and Links

Solutions .
Introduction to Polynomials
Graphs of Polynomial Functions
Middle School Math (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers
Primary Math (Grades 4 and 5) with Free Questions and Problems With Answers