6.3 Simplifying algebraic expressions
Most algebraic expressions contain multiple terms. Some are like terms, some are constants and some are unlike terms.
Terms are separated by addition and subtraction symbols, and joined into one by multiplication symbols, division symbols, and brackets. Therefore, look for addition and subtractions symbols to distinguish one term from the next.
Worked example 6.6: Terms, coefficients and variables
Look the expression below and answer the questions that follow:
\[4x^{2} + 1 - 5y^{2} + 3b^{2} + 2f^{4}\]- How many terms are there in the expression?
- What is the value of the coefficient of \(y^{2}\)?
- What is the value of the constant term?
Determine the number of terms in the expression.
\[4x^{2} + 1 - 5y^{2} + 3b^{2} + 2f^{4}\]Hint: The terms are separated by addition and subtraction symbols.
The terms in the expression are \(4x^{2}\), \(+ 1\), \(- 5y^{2}\), \(+ 3b^{2}\), and \(+ 2f^{4}\).
The expression therefore has \(5\) terms.
Determine the value of the coefficient of \(y^{2}\).
Hint: Remember, the coefficient of a variable is the number (or letter) that is multiplied by that variable.
The term containing \(y^{2}\) is \(- 5y^{2}\). Therefore the coefficient of \(y^{2}\) is \(−5\).
Hint: You must always include the sign of the term in the coefficient. The answer is \(−5\) and not \(5\).
Determine the value of the constant term.
The constant term is the term in an expression that does not contain any variables.
The only term in the expression without any variables is \(+1\). Therefore, the constant term is \(+1\).
When there are like terms in an algebraic expression, it is possible to simplify the expression to reduce the number of terms. We can simplify the expression by collecting like terms.
Worked example 6.7: Collecting like terms
Simplify the expression: \(7x - 3x = 4x\).
Identify the coefficients of the terms to the left of the equals sign.
The expression is made of the terms, \(7x\) and \(- 3x\).
The coefficient of the first term, \(7x\), is \(7\).
The second term is \(- 3x\). Notice that the second term has a minus sign in front of it. That means that the coefficient of the second term is negative. This is because the expression \(7x - 3x\) is the same as \(7x + ( - 3)x\). So the coefficient of the second term is \(−3\).
Identify the coefficient of the answer.
The question tells us that \(7x - 3x\) is equal to \(4x\). The second part of the question asks for the coefficient of the answer, which is \(4x\).
The coefficient of the answer, \(4x\), is \(4\).
Worked example 6.8: Simplifying algebraic expressions
Simplify the following expression: \(7x + 4x^{2} + 5 + 3x^{2}\).
Identify all the terms in the expression.
\(7x + 4x^{2} + 5 + 3x^{2}\) has four terms: \(7x\), \(+ 4x^{2}\), \(+ 5\) and \(+ 3x^{2}\).
Collect like terms.
We can only add like terms. Like terms have the same variables and the same exponents.
\[7x + \mathbf{4}\mathbf{x}^{\mathbf{2}} + 5 + \mathbf{3}\mathbf{x}^{\mathbf{2}}\] \[= \mathbf{7}\mathbf{x}^{\mathbf{2}} + 7x + 5\]Hint: We usually write expressions from the highest power (\(x^{2}\)) to the lowest power, or the constant. You don’t have to do this, but it makes the expression a bit easier to read.
Worked example 6.9: Simplifying algebraic expressions
Simplify the following expression: \(7xy + 6x - 4xy - 3xy\).
Identify all the terms in the expression.
\(7xy + 6x - 4xy - 3xy\) has four terms: \(7xy\), \(+ 6x\), \(- 4xy\), \(- 3xy\).
Identify the like terms and add them together
We can only add like terms. Like terms have the same variables and the same exponents.
\[7xy + 6x - 4xy - 3xy\] \[= \left( 7xy - 4xy - 3xy \right) + 6x\] \[= 6x\]You might think that \(7xy - 4xy - 3xy = 0xy\). But that is not fully simplified, because:
\[0xy = 0 \times xy\]When we multiply a number by zero, we get zero.
Therefore, \(0xy = 0\)
\[0 + 6x = 6x\]So the final answer is \(6x\).
Worked example 6.10: Simplifying algebraic expressions
Simplify the following expression by collecting like terms:
\[8x^{2}y + 8y + 9xy^{2} + 4x^{2}y\]Identify all the terms in the expression.
\(8x^{2}y + 8y + 9xy^{2} + 4x^{2}y\) has four terms: \(8x^{2}y\), \(+ 8y\), \(+ 9xy^{2}\) and \(+ 4x^{2}y\).
Collect the like terms.
We can only add like terms. Like terms have the same variables and the same exponents.
\[8x^{2}y + 8y + 9xy^{2} + 4x^{2}y\] \[= 12x^{2}y + 9xy^{2} + 8y\]In \(x^{2}y\), the \(x\) is squared, but the \(y\) is not. In \(xy^{2}\), the \(y\) is squared, but the \(x\) is not. So, \(x^{2}y\) and \(xy^{2}\) are not like terms.