Question

How do you factor polynomials by finding the greatest common factor?

Answer 1

Hint: In this question, we have to find the factor of a polynomial. So, we will apply the greatest common factor method to get the solution. As we know, a polynomial is an equation which consists of some variables and numbers. A polynomial is of three types; monomial, binomial and the trinomial. Thus, we will find the factors of each polynomial by finding the factors of each term in every polynomial. Then, we will take the greatest common factor from each term. In the last, we will multiply all the GCD of each polynomial and group the remaining inside the brackets, to get the solution for the problem.

Complete step by step solution:
According to the question, we have to find the factors of a polynomial. As we know, polynomials are of three types that are monomial, binomial and trinomial. Thus, we will solve all the three polynomials using the greatest common factor.
The monomial polynomial is that kind of polynomial which has only one term. Let us suppose the monomial polynomial is $f(x)=3{{x}^{2}}$ ------- (1)
Now, we will find the factors of the constant and variable, we get
$\begin{align}
  & 3=1\times 2 \\
 & {{x}^{2}}=x\times x \\
\end{align}$
Since, we see no term is common in both the variable and the constant, therefore the factors of $f(x)=3{{x}^{2}}$ is equal to $3{{x}^{2}}$ .
Now, we know that, a binomial polynomial consists of two terms, thus let us say $g(x,y)=4{{x}^{3}}+2xy$ is a binomial polynomial.
Now, we will find the factors of each variable and constant of g(x,y) function, we get
$\begin{align}
  & 4=2\times 2 \\
 & {{x}^{3}}=x\times x\times x \\
 & 2=2 \\
 & x=x \\
 & y=y \\
\end{align}$
Thus, the function will become $g(x,y)=2\times 2\times x\times x\times x+2\times x\times y$ ----- (1)
Therefore, we see the common numbers from the constants is 2 and the common variable from both the variables is x.
Thus, we will take 2x common from the function (1) and the rest will be written under the brackets, we get
$g(x,y)=2x\left( 2\times x\times x+y \right)$
On further simplification, we get
$g(x,y)=2x\left( 2{{x}^{2}}+y \right)$
Therefore, the factors of the binomial polynomial $g(x,y)=4{{x}^{3}}+2xy$ is equal to $g(x,y)=2x\left( 2{{x}^{2}}+y \right)$ .
Now, a trinomial polynomial consists of three terms, thus let us say $h(x,y)=4{{x}^{3}}+2{{x}^{2}}y+16{{x}^{4}}{{y}^{2}}$ is a trinomial polynomial.
Now, we will find the factors of each variable and constant of h(x,y) function, we get
$\begin{align}
  & 4=2\times 2 \\
 & {{x}^{3}}=x\times x\times x \\
 & 2=2 \\
 & x=x\times x \\
 & y=y \\
 & 16=2\times 2\times 2\times 2 \\
 & {{x}^{4}}=x\times x\times x\times x \\
 & {{y}^{2}}=y\times y \\
\end{align}$
Thus, the function will become $h(x,y)=2\times 2\times x\times x\times x+2\times x\times x\times y+2\times 2\times 2\times 2\times x\times x\times x\times x\times y\times y$ ----- (2)
Therefore, we see the common numbers from the constants is 2 and the common variable from both the variables is ${{x}^{2}}$ .
Thus, we will take $2{{x}^{2}}$ common from the function (2) and the rest will be written under the brackets, we get
\[h(x,y)=2{{x}^{2}}\left( 2\times x+y+2\times 2\times 2\times x\times x\times y\times y \right)\]
On further solving the above equation, we get
\[h(x,y)=2{{x}^{2}}\left( 2x+y+8{{x}^{2}}{{y}^{2}} \right)\]
Therefore, the factors of the trinomial polynomial $h(x,y)=4{{x}^{3}}+2{{x}^{2}}y+16{{x}^{4}}{{y}^{2}}$ is equal to \[h(x,y)=2{{x}^{2}}\left( 2x+y+8{{x}^{2}}{{y}^{2}} \right)\] .
Thus, we find the factors of each polynomial using the greatest common factor method.

Note: While solving the question, always remember that the polynomial is distinguish through the number of terms in the polynomial. Also, after finding the factors of each term, take the greatest common factor among all.