Question

How do you use the distributive property to factor $21b - 15a$?

Answer 1

Hint: To factor a polynomial, first identify the greatest common factor of the monomial terms. Then use the distributive property to rewrite the polynomial as the product of the GCF and the other parts of the polynomial.

Formula used:
Distributive property of multiplication over subtraction:
Let $a$, $b$ and $c$ be three real numbers, then
 $a \times (b - c) = (a \times b) - (a \times c)$

Complete step by step solution:
Remember that factoring is the process of breaking a number down into its multiplicative components. Try to identify factors that the terms of the polynomial have in common.
To factor a polynomial, first identify the greatest common factor of the monomial terms. Then use the distributive property to rewrite the polynomial as the product of the GCF and the other parts of the polynomial.
We know that both terms in a polynomial are divisible by their GCF, so we can rewrite each polynomial as a product of the GCF and the combined "leftover" factors of each monomial.
 Let's go through the process of factoring a polynomial, step by step.
$21b - 15a$ is a polynomial.
$21b$ and $ - 15a$ are monomials.
Factor $21b$.
$3 \cdot 7 \cdot b$
Factor $ - 15a$.
$ - 1 \cdot 3 \cdot 5 \cdot a$
Find the GCF.
$3$
Rewrite each monomial with the GCF as one factor.
$21b = 3 \cdot 7b$
$ - 15a = 3 \cdot \left( { - 5a} \right)$
Rewrite the polynomial expression using the factored monomials in place of the original terms.
$3\left( {7b} \right) - 3\left( {5a} \right)$
Use the distributive property to pull out the GCF.
$3\left( {7b - 5a} \right)$

Final solution: Hence, $21b - 15a$ can be factored as $3\left( {7b - 5a} \right)$.

Note: To reassure ourselves that this is correct, we can multiply $3 \cdot \left( {7b - 5a} \right)$, checking to see if we get the original form of the polynomial, $21b - 15a$.
 The factored form of a polynomial is one in which the polynomial is written as a product of factors, and each non-monomial factor has no common factors in its terms. For example, the factors of $3\left( {7b - 5a} \right)$ are $3$ and $7b - 5a$. The two terms of $7b - 5a$ have no common factors.