Question

What are Monomial factors of a Polynomial?

Answer 1

Hint: To understand the term monomial factors, we must individually understand the two terms monomial and factors. Using our understanding of this concept, we can say that the terms which are common in every single term of the polynomial is the monomial factor of that polynomial.

Complete step-by-step solution:
We know that for any expression in x, a monomial is an expression that is the product of constants and non-negative integer powers of x. Hence, we can also say that a polynomial is the sum of monomials.
We know that a polynomial could be of multiple types, like monomial, binomial, etc. We are very well aware that a polynomial is that expression or equation which includes variables and coefficients and has one or more than one term. Thus, each term of a polynomial is a monomial.
So, we can say that, when a polynomial is expressed in the factor form, or when the polynomial is factored, each of the monomial terms are called the monomial factors for that polynomial.
In other words, we can say that the monomial factors of a polynomial are those terms that appear in every single term of that polynomial.
Let us understand the monomial factors of a polynomial with the help of an example.
Let us assume a polynomial $y=3{{x}^{7}}+6{{x}^{6}}+9{{x}^{5}}+18{{x}^{4}}$.
We can simplify this polynomial as follows,
$y=3{{x}^{4}}\left( {{x}^{3}}+2{{x}^{2}}+3x+6 \right)$
Now, we can further simplify this,
$y=3{{x}^{4}}\left\{ {{x}^{2}}\left( x+2 \right)+3\left( x+2 \right) \right\}$
$\Rightarrow y=3{{x}^{4}}\left( x+2 \right)\left( {{x}^{2}}+3 \right)$
So, now we can see that in this factored form of the polynomial, the monomial term is $3{{x}^{4}}$.
Hence, $3{{x}^{4}}$ is the monomial factor of this polynomial.

Note: Here, in this example, we must note that $3{{x}^{7}},6{{x}^{6}},9{{x}^{5}}\text{ and }18{{x}^{4}}$ are monomial terms, but these are not monomial factors. Also, in the factored form, we can see that $3{{x}^{4}},\left( x+2 \right)\text{ and }\left( {{x}^{2}}+3 \right)$ are the factors, but only the term $3{{x}^{4}}$ is the monomial factor.