Question

How do you factor the monomial $95x{y^2}$ completely?

Answer 1

Hint: First, we have to write out the prime factorization of the coefficient, i.e., to take prime factors of the coefficient, $95$ . Then, we write down all the variables out, instead of using exponents. We will get the completely factored monomial of $95x{y^2}$ .

Complete step-by-step solution:
We know that factoring is a process of breaking down a monomial into smaller terms. Although factoring an equation usually shortens the equation, factoring a polynomial expands it.
Let’s review the definition of a monomial first.
Monomial: An algebraic expression consisting of one term. Some monomials might have more than one variable.
When we factor a monomial, we are breaking it down into smaller terms. We have to express the monomial as a product of smaller monomials.
Completely factoring, or completely expanding, a monomial is a simple method. First, we determine the prime factorization of the coefficient. Then, we write down all the variables out, instead of using exponents.
Now, we look at how to factor $95x{y^2}$ .
First step is to take prime factorization of the coefficient, $95$ . We can write this as $1 \times 95$ .
$95 = 1 \times 95$
Then, we can break $95$ into $5$ and $19$ .
$95 = 1 \times 5 \times 19$
Since, $5$ and $19$ are prime, we can stop here.
So, the prime factorization of $95$ is:
$95 = 1 \times 5 \times 19$
Second step is to write all the variables without exponents.
$x{y^2} = x \cdot y \cdot y$

Finally, we can write the whole thing out. The factor of $95x{y^2}$ is:
$95x{y^2} = 1 \cdot 5 \cdot 19 \cdot x \cdot y \cdot y$

Note: Factoring a monomial, an algebraic expression consisting of one term is not the same as completely factoring it. Completely factoring a monomial is writing down its entire prime factorization of the coefficient and all the variables instead of using exponents. Factoring a monomial is more general. It means breaking down the monomial into smaller terms. These smaller terms in the monomial do not have to be completely factored themselves. For example:
Monomial: $95x{y^2}$
Completely factored monomial:
$1 \cdot 5 \cdot 19 \cdot x \cdot y \cdot y$
Factored monomial:
$1 \cdot 95 \cdot x \cdot y \cdot y$
Notice that each part of the completely factored monomial, $1 \cdot 5 \cdot 19 \cdot x \cdot y \cdot y$ , is completely factored. For example, you can’t factor $5$ or $y$ any more than it already is.
Also, we can notice that each part of the factored monomial, $1 \cdot 95 \cdot x \cdot y \cdot y$ , is not necessarily completely factored. For example, $x$ is completely factored, but $95$ is not.