Question

How do you factor $7x+28$ ? \[\]

Answer 1

Hint: We recall the Euclidean division of polynomials and where we say the divisor polynomial is a factor polynomial when the remainder polynomial is zero. We take 7 common from both the terms of the given polynomial $7x+28$ and factorize. We check if the factors can be factored further. If we cannot factorize further than we have factored completely.

Complete step by step answer:
We know that when we divide a dividend polynomial $p\left( x \right)$ with degree $n$ by some divisor polynomial $d\left( x \right)$ with degree $m\le n$ then we get the quotient polynomial $q\left( x \right)$ of degree $n-m$ and the remainder polynomial as $r\left( x \right)$.We use Euclidean division formula and can write as
\[ p\left( x \right)=d\left( x \right)q\left( x \right)+r\left( x \right)\]
We also know that if the remainder polynomial is zero then we call $d\left( x \right),q\left( x \right)$ factor polynomial of $p\left( x \right)$ or simply factors of $p\left( x \right)$. If ${{p}_{1}}\left( x \right),{{p}_{2}}\left( x \right),...,{{p}_{k}}\left( x \right)$ are $k$ factors of $p\left( x \right)$ then we say $p\left( x \right)={{p}_{1}}\left( x \right){{p}_{2}}\left( x \right)...{{p}_{k}}\left( x \right)$ is factored completely if none of the factors ${{p}_{1}}\left( x \right),{{p}_{2}}\left( x \right),...,{{p}_{k}}\left( x \right)$ can be factored further. We are given the following polynomial in the question
\[7x+28\]
We can write the given polynomial as
\[7x+28=7\cdot x+7\cdot 4\]
 We see that 7 is a factor of both the terms in the given polynomial. We use distributive property of multiplication over addition and take 7 common to have;
\[7x+49=7\cdot x+7\cdot 4=7\left( x+4 \right)\]

Here we have obtained factors 7 and $x+4$. Since we cannot factorize either of 7 or $x+4$ further, we have factored completely.

Note: We note that the highest power on the variable is called degree of the polynomial. If degree is 1 we call the polynomial a linear polynomial. If the polynomial is a single term then we call the polynomial a monomial and if it has two terms it is called binomial. Here the given polynomial $7x+28$has been factored into monomial constant 7 and a linear binomial $x+4$. The values of $x$ for which the polynomial is zero we call zeros or roots of the polynomial. Here $x=-4$ is zero of the polynomial $7x+28$.