Distributing and Factoring
Distributing and factoring are two of the most important techniques in algebra. They give you ways of manipulating expressions without changing the expression’s value. So it follows that you can factor or distribute one side of the equation without doing the same for the other side of the equation.
The basis for both techniques is the following property, called the distributive property:
Similarly:
a can be any kind of term, from a variable to a constant to a combination of the two.
Distributing
When you distribute a factor into an expression within parentheses, you simply multiply each term inside the parentheses by the factor outside the parentheses. For example, consider the expression 3y(y2 – 6):
If we set the original, undistributed expression equal to another expression, you can see why distributing facilitates the solving of some equations. Solving 3y (y2 – 6) = 3y3 + 36 looks quite difficult. But if you distribute the 3y, you get:
Subtracting 3y3 from both sides gives us:
Factoring
Factoring an expression is essentially the opposite of distributing. Consider the expression 4x3 – 8x2 + 4x, for example. You can factor out the GCF of the terms, which is 4x:
The expression simplifies further:
See how useful these techniques are? You can group or ungroup quantities in an equation to make your calculations easier. In the last example from the previous section on manipulating equations, we distributed andfactored to solve an equation. First, we distributed the quantity log 3 into the sum of x and 2 (on the right side of the equation). We later factored the term x out of the expression x log 2 – x log 3 (on the left side of the equation).
Distributing eliminates parentheses, and factoring creates them. It’s your job as a Math IC mathematician to decide which technique will best help you solve a problem.
Combining Like Terms
After factoring and distributing, there are additional steps you can take to simplify expressions or equations. Combining like terms is one of the simpler techniques you can use, and involves adding or subtracting the coefficients of variables that are raised to the same power. For example, by combining like terms, the expression:
can be simplified to:
by adding the coefficients of the variable x3 together and the coefficients of x2 together.
Generally speaking, when you have an expression in which one variable is raised to the same power in different terms, you can factor out the variable and add or subtract the coefficients, combining them into one coefficient and therefore combining the “like” terms into one term. A general formula for combining like pairs looks something like this:
