Grade 9 examples on addition, subtraction, mltiplication, division and simplification of monomilas are presented along with their detailed solutions. More questions and their solutions and detailed explanations are included.
A monomial is the product of a real number and variables raised to non negative integer powers (exponents).
Example 1
Take a number, \( - 2 \) for example, and a variable with exponents equal to \( 2 \), \( x^2 \) for example, and multiply them to obtain
\( \quad - 2 \times x^2 \)
Now, for simplicity, take the multiplication symbol out and write it as
\(\quad - 2 x^2 \) to obtain a monomial.
These are examples of monomials
\( \quad x , 2 x , 3x^2 , - 0.1 x , - \dfrac{3}{4} x^2 y^2 , - y \)
The coefficient of a monomial is the number part ( that is at the front ) of the monomial.
Example 2
The coefficient of the monomial \( \color{red}{2} x \) is \( \color{red}{2} \)
The coefficient of the monomial \( \color{red}{- 3} x^2 \) is \( \color{red}{-3} \)
The coefficient of the monomial \( \color{red}{- \dfrac{5}{2}} y^3 x \) is \( \color{red}{- \dfrac{5}{2}} \)
The coefficient of the monomial \( x = \color{red}{1} x\) is \( \color{red}{1} \) NOTE that it is not written \( 1 x \) it is instead written as \( x \) for simplicity
The coefficient of the monomial \( - x^2 = \color{red}{-1} x^2 \) is \( \color{red}{- 1} \) NOTE that it is not written \( - 1 x^2 \) it is instead written as \( - x^2 \) for simplicity
We can only add and subtract monomials with like terms that have the same variables to the same power.
Examples of monomials with like terms
\( \quad - 3 x^2 , 4 x^2 , - x^2 \) are all monomials with like terms \( x^2 \) and may be added
\( \quad - y^2 x , 4 y^2 x , - x y^2 \) are all monomials with like terms \( x y^2 \) and may be added.
NOTE that the terms \( x y^2 \) and \( y^2 x \) are the same
Example 3
Add and/or subtract the following monomials
a) \(2 x + 4 x\) b) \( 3 x^2 - x^2 \) c) \( x y - 5 x y \) d) \( 3 x^2 y - x^2 y + 4 y x^2 \) e) \( - x + x \)
Solution to Example 3
a)
\( \quad \quad \begin{split}
\color{red}{2}x + \color{red}{4} x & = \color{red}{(2 + 4)} x \quad \quad \text{factor variable \( x \) out and put all coefficients between brackets}\\\\
& = 6 x \quad \quad \text{add coefficients between brackets}
\end{split} \)
b)
\( \quad \quad \begin{split}
\color{red}{3} x^2 \color{red}{-1} x^2& = \color{red}{(3 - 1)} x^2 \quad \quad \text{factor variable \( x^2 \) out and put all coefficients between brackets}\\\\
& = 2 x^2 \quad \quad \text{add/subtract coefficients between brackets}
\end{split} \)
c)
\( \quad \quad \begin{split}
\color{red}{1} x y \color{red}{- 5} x y & = \color{red}{(1 - 5)} x y \quad \quad \text{factor variable \( x y \) out and put all coefficients between brackets}\\\\
& = - 4 x y \quad \quad \text{add/subtract coefficients between brackets}
\end{split} \)
d)
\( \quad \quad \begin{split}
3 x^2 y - x^2 y + 4 y x^2 & = \color{red}{(3-1+4)} x^2 y \quad \quad \text{factor variable \( x^2 y \) out and put all coefficients between brackets}\\\\
& = 6 x^2 y \quad \quad \text{add/subtract coefficients between brackets}
\end{split} \)
NOTE that the above example part d) terms \( x^2 y \) and \( y x^2\) are the same
e)
\( \quad \quad \begin{split}
- x + x & = (-1 + 1) x \quad \quad \text{factor variable \( x \) out and put all coefficients between brackets}\\\\
& = (-1+1)x = 0 x = 0 \quad \quad \text{add/subtract coefficients between brackets}
\end{split} \)
The following rule of exponents is widely used in the multiplication of exponent forms
\( x^m \cdot x^n = x^{m+n} \)
You can multiply any two monomials and they DO NOT have to have like terms.
Example 4
Multiply the following monomials
a) \( ( x) (6 x) \) b) \( (3 x^2) (-2 x) \) c) \( (5 x^2 y) (- y^2 x) \) d) \( (\dfrac{2}{3} x y) (- \dfrac{5}{4} y z) \) d) \( ( 6 ) (- y z) \)
Solution to Example 4
a)
\( \quad \quad \begin{split}
( x) (6 x) & = (\color{red}{1} \color{blue}{x} )(\color{red}{6} \color{blue}{x}) \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = \color{red}{(1 \cdot 6)} \color{blue}{(x \cdot x)} \quad \quad \text{Multiply the coefficients together and the terms with the same variable together} \\\\
& = 6 x^{1+1} = 6 x^2 \quad \quad \text{Evaluate the multiplication of the coefficients and multiply variables using the rule of exponents above} \\\
\end{split} \)
b)
\( \quad \quad \begin{split}
(3 x^2) (-2 x) & = (\color{red}{3} \color{blue}{x^2} )(\color{red}{(-2)} \color{blue}{x}) \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = \color{red}{(3 \cdot (-2))} \color{blue}{(x^2 \cdot x)} \quad \quad \text{Multiply the coefficients together and the terms with the same variable together} \\\\
& = - 6 x^{2+1} = - 6 x^3 \quad \quad \text{Evaluate the multiplication of the coefficients and multiply variables using the rules of exponents above} \\\
\end{split} \)
c)
\( \quad \quad \begin{split}
(5 x^2 y) (- y^2 x) & = (5 x^2 y )((-1) y^2 x ) \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = (5 \cdot (-1))(x^2 \cdot x) (y^2 \cdot y) \quad \quad \text{Multiply the coefficients together and the terms with the same variable together} \\\\
& = - 5 y^{2+1} x^{2+1} = - 5 y^3 x^3 \quad \quad \text{Evaluate the multiplication of the coefficients and multiply variables using the rules of exponents} \\\
\end{split} \)
d)
\( \quad \quad \begin{split}
( \dfrac{2}{3} x y) (- \dfrac{5}{4} y z) & = ((\dfrac{2}{3}) x y )((-\dfrac{5}{4}) y z ) \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = (\dfrac{2}{3} \cdot (-\dfrac{5}{4}))(x) (y \cdot y) (z) \quad \quad \text{Multiply the coefficients together and the terms with the same variable together} \\\\
& = - \dfrac{5}{6} x y^{1+1} z = - \dfrac{5}{6} x y^2 z \quad \quad \text{Evaluate the multiplication of the coefficients and multiply variables using the rules of exponents} \\\
\end{split} \)
e)
\( \quad \quad \begin{split}
( 6 ) (- y z) & = ( 6 ) ((-1) y z) \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = (6 \cdot (-1)) yz \quad \quad \text{Multiply the coefficients } \\\\
& = - 6 yz \quad \quad \text{Evaluate the multiplication of the coefficients}\\\
\end{split} \)
The following rule of exponents is widely used in the division of exponent forms
\( \dfrac{x^m}{x^n} = x^{m-n} \)
You can divide any two monomials and they DO NOT have to have like terms. However the divisor must not be equal to zero.
In the example and the exercises below, we assume that NONE of the variables is equal to zero.
Example 5
Divide the following monomials
a) \( \dfrac{- x^2}{ x} \) b) \( \dfrac{4 x^4}{ x^2} \) c) \( \dfrac{- 2 x^4 y^3}{ 6 x^2 y^2} \) d) \( \dfrac{- 12 x y z^3}{ 6 x y } \) e) \( \dfrac{- 12 x^2y^3}{ -3} \)
Solution to Example 5
a)
\( \quad \quad \begin{split}
\dfrac{- x^2}{ x} & = \dfrac{ \color{red}{- 1} \color{blue}{x^2}}{ \color{red}{1} \color{blue}{x }} \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = \color{red}{\dfrac{-1}{1}} \cdot \color{blue}{\dfrac{x^2}{x}} \quad \quad \text{Divide the coefficients and the terms with the same variable separately} \\\\
& = - 1 \cdot x^{2-1} = - 1 \cdot x^{1} = - x \quad \quad \text{Evaluate the division of the coefficients and simplify variables using the rules of division of exponents above} \\\
\end{split} \)
b)
\( \quad \quad \begin{split}
\dfrac{4 x^4}{ x^2} & = \dfrac{ \color{red}{4} \color{blue}{x^4}}{ \color{red}{1} \color{blue}{x^2}} \quad \quad \text{identify, if necessary, the coefficients and the variables of each monomial}\\\\
& = \color{red}{ \dfrac{4}{1} } \cdot \color{blue}{ \dfrac{x^4}{x^2} } \quad \quad \text{Divide the coefficients and the terms with the same variable separately} \\\\
& = 4 \cdot x^{4-2} = 4 x^{2} \quad \quad \text{Evaluate the division of the coefficients and simplify variables using the rules of division of exponents above} \\\
\end{split} \)
c)
\( \quad \quad \begin{split}
\dfrac{- 2 x^4 y^3}{ 6 x^2 y^2} & = \color{red}{ \dfrac{-2}{6}} \color{blue}{\cdot \dfrac{x^4}{x^2}} \color{green}{ \cdot \dfrac{y^3}{y^2}} \quad \quad \text{Divide the coefficients and the terms with the same variable separately} \\\\
& = - \dfrac{1}{3} \cdot x^{4-2} \cdot y^{3-1} = - \dfrac{1}{3} x^{2} y \quad \quad \text{Evaluate the division of the coefficients and simplify variables using the rules of division of exponents} \\\
\end{split} \)
d)
\( \quad \quad \begin{split}
\dfrac{- 12 x y z^3}{ 6 x y } & = \dfrac{-12}{6} \cdot \dfrac{x}{x} \cdot \dfrac{y}{y} \cdot z^3\quad \quad \text{Divide the coefficients and the terms with the same variable separately} \\\\
& = - 2 \cdot x^{1-1} \cdot y^{1-1} \cdot z^3 = - 2 z^3 \quad \quad \text{Evaluate the division of the coefficients and simplify variables using the rules of division of exponents } \\\
\end{split} \)
e)
\( \quad \quad \begin{split}
\dfrac{- 12 x^2y^3}{ -3} & = \dfrac{-12}{-3} \cdot x^2y^3 \quad \quad \text{Divide the coefficients} \\\\
& = 4 x^2y^3 \quad \quad \text{Evaluate the division of the coefficients} \\\
\end{split} \)
