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
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.