Question

How do you factor completely $7x+49$ ? \[\]

Answer 1

Hint: We recall the Euclidean division of polynomials and where we say the divisor polynomial is a factored polynomial when the remainder polynomials. We take 7 common from both the terms of the given polynomial $7x+49$ and factorize. We check if the factors can be factored further. If we cannot factorize further then we have factored completely. \[\]

Complete step-by-step solution:
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 $r\left( x \right)$ of degree either equal to $m$ or $m-1$. 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
\[7x+49\]
We can write the given polynomial as
\[7x+49=7\cdot x+7\cdot 7\]
 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;
\[7x+49=7\cdot x+7\cdot 7=7\left( x+7 \right)\]
Here we have obtained factors 7 and $x+7$. Since we cannot factorized wither of 7 or $x+7$ we have factored completely.\[\]

Note: We note that the highest power on the variable is called the degree of the polynomial. If the degree is zero then the polynomial is a constant and if the degree is 1 we call the polynomial a linear polynomial. Here the given polynomial $7x+49$ has been factored into constant 7 and a linear polynomial $x+7$. The values of $x$ for which the polynomial is zero we call zeros or roots of the polynomial. Here $x=-7$ is zero of the polynomial $7x+49$.