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.