This lesson is designed for Grade 9 students to master operations involving monomials. A monomial is a single algebraic term consisting of a real number multiplied by variables raised to non-negative integer powers.
Below, you will find clear examples demonstrating how to identify coefficients, as well as how to add, subtract, multiply, and divide monomials. Once you review the examples, test your understanding with the practice questions, which include interactive step-by-step solutions.
1. Understanding the Coefficients of Monomials
Take a number, such as \(-2\), and a variable with an exponent, such as \(x^2\), and multiply them together: \(-2 \times x^2\). For simplicity, we remove the multiplication symbol and write it as a monomial: \(-2x^2\).
Examples of standard monomials include: \( x \), \( 2x \), \( 3x^2 \), \( -0.1x \), \( -\dfrac{3}{4}x^2y^2 \), \( -y \).
Definition: Coefficient
The coefficient of a monomial is the numerical part located at the front of the term.
Example 1: Identifying Coefficients
- The coefficient of \( \color{red}{2}x \) is \(2\).
- The coefficient of \( \color{red}{-3}x^2 \) is \(-3\).
- The coefficient of \( \color{red}{-\dfrac{5}{2}}y^3x \) is \(-\dfrac{5}{2}\).
- The coefficient of \( x \) (which is \( \color{red}{1}x \)) is \(1\). Note: We write \(x\), not \(1x\).
- The coefficient of \( -x^2 \) (which is \( \color{red}{-1}x^2 \)) is \(-1\). Note: We write \(-x^2\), not \(-1x^2\).
2. Add and Subtract Monomials
We can only add and subtract monomials that are like terms (meaning they have the exact same variables raised to the exact same powers).
- \( -3x^2 \), \( 4x^2 \), and \( -x^2 \) are like terms and can be added.
- \( -y^2x \), \( 4y^2x \), and \( -xy^2 \) are like terms and can be added. (Note: \(xy^2\) and \(y^2x\) are identical).
Example 2: Addition and Subtraction
a) \( 2x + 4x \)
\[ \begin{aligned} \color{red}{2}x + \color{red}{4}x &= \color{red}{(2 + 4)}x \quad && \text{(factor out the variable } x\text{)} \\ &= 6x \quad && \text{(add the coefficients)} \end{aligned} \]
b) \( 3x^2 - x^2 \)
\[ \begin{aligned} \color{red}{3}x^2 \color{red}{- 1}x^2 &= \color{red}{(3 - 1)}x^2 \quad && \text{(factor out the variable } x^2\text{)} \\ &= 2x^2 \quad && \text{(subtract the coefficients)} \end{aligned} \]
c) \( 3x^2y - x^2y + 4yx^2 \)
\[ \begin{aligned} 3x^2y - 1x^2y + 4x^2y &= \color{red}{(3 - 1 + 4)}x^2y \quad && \text{(group the coefficients)} \\ &= 6x^2y \end{aligned} \]
3. Multiply Monomials
Multiplication Rule of Exponents
You can multiply any two monomials; they do not need to be like terms. Use the product rule for exponents:
\( x^m \cdot x^n = x^{m+n} \)
Example 3: Multiplication
a) \( (x)(6x) \)
\[ \begin{aligned} (\color{red}{1}\color{blue}{x})(\color{red}{6}\color{blue}{x}) &= \color{red}{(1 \cdot 6)}\color{blue}{(x \cdot x)} \quad && \text{(group coefficients and variables)} \\ &= 6x^{1+1} = 6x^2 \quad && \text{(apply exponent rule)} \end{aligned} \]
b) \( (5x^2y)(-y^2x) \)
\[ \begin{aligned} (5x^2y)(-1y^2x) &= (5 \cdot -1)(x^2 \cdot x)(y \cdot y^2) \\ &= -5 x^{2+1} y^{1+2} \\ &= -5x^3y^3 \end{aligned} \]
4. Divide Monomials
Division Rule of Exponents
You can divide any two monomials as long as the divisor is not zero. Use the quotient rule for exponents:
\( \dfrac{x^m}{x^n} = x^{m-n} \)
Example 4: Division
a) \( \dfrac{-x^2}{x} \)
\[ \begin{aligned} \dfrac{\color{red}{-1}\color{blue}{x^2}}{\color{red}{1}\color{blue}{x}} &= \color{red}{\dfrac{-1}{1}} \cdot \color{blue}{\dfrac{x^2}{x}} \\ &= -1 \cdot x^{2-1} \\ &= -x \end{aligned} \]
b) \( \dfrac{-2x^4y^3}{6x^2y^2} \)
\[ \begin{aligned} \dfrac{-2x^4y^3}{6x^2y^2} &= \color{red}{\dfrac{-2}{6}} \cdot \color{blue}{\dfrac{x^4}{x^2}} \cdot \color{green}{\dfrac{y^3}{y^2}} \\ &= -\dfrac{1}{3} \cdot x^{4-2} \cdot y^{3-2} \\ &= -\dfrac{1}{3}x^2y \end{aligned} \]