audio read-through Factoring Simple Expressions

Need some basic practice recognizing products and sums? Recognizing Products and Sums; Identifying Factors and Terms

Need some basic practice identifying common factors? Identifying Common Factors

DEFINITION: to factor an expression
To factor an expression means to take the expression and rename it as a product.

That is, to factor an expression means to write the expression as a product.

Examples

Question: Factor: $\, ab + ac$
Solution: $ab + ac = a(b + c)$

The expression $\,ab + ac\,$ is a sum, since the last operation is addition. The expression $\,a(b + c)\,$ is a product, since the last operation is multiplication. The process of factoring took us from the sum $\,ab + ac\,$ to the product $\,a(b + c)\,.$

Notice that $\,\,ab + ac = a(b + c)\,\,$ is just the distributive law, backwards!

In going from the name $\,ab + ac\,$ to the name $\, a(b + c) \,,$ the common factor ($\,a\,$) is first identified, and written down. Next, an opening parenthesis ‘ ( ’ is inserted. Then, the remaining parts of each term are written down. Finally, the closing parenthesis ‘ ) ’ is inserted.

Question: Write in factored form: $\,3x - 3t\,$
Solution: $3(x - t)$
Question: Write in factored form: $\,2xy - 2yz$
Solution: $2y(x - z)$
Question: Write in factored form: $\,5x^2 - x^2y^2$
Solution: $x^2(5 - y^2)$

Note: In the exercises below, exponents are typed in using the ‘^’ key. For example, $\, x^2(5 - y^2) \,$ is typed in as:   x^2(5 - y^2)

Question: Write in factored form: $\,x(2x + 1) - 3(2x + 1)$
Solution: $(2x + 1)(x - 3)$

Note: The product $\,(2x+1)(x-3)\,$ can also be written as $\,(x-3)(2x+1)\,.$ There is no convention here about which name is ‘best’. The exercise below recognizes both answers.

Practice