Question

Examine whether $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$

Answer 1

Hint:
We can divide the given polynomial with $x + 2$ using a long division method. Then we can check the remainder. If the remainder is zero, $x + 2$ will be a factor of the given polynomial. If the remainder is non zero, then, $x + 2$ will not be a factor of the given polynomial.

Complete step by step solution:
We need to check whether $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$ .
So, we can try dividing ${x^3} + 3{x^2} + 5x + 6$ with $x + 2$ .
If the remainder is zero, we can say that the ${x^3} + 3{x^2} + 5x + 6$ is divisible by $x + 2$ and the term $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$ .
If the remainder is not zero, we can say that the ${x^3} + 3{x^2} + 5x + 6$ is not divisible by $x + 2$ and the given term $x + 2$ is not a factor of ${x^3} + 3{x^2} + 5x + 6$ .
So, by dividing, we get,

So, after the division, we get the remainder as zero. So, we can say that ${x^3} + 3{x^2} + 5x + 6$ is divisible by $x + 2$ .

Therefore, $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$.

Note:
Alternate solution to the problem is given by,
If $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$ , then the root of the factor will also be a root of the polynomial.
So, we can find the roots of the expression $x + 2$ by equating it to zero.
 $ \Rightarrow x + 2 = 0$
 $ \Rightarrow x = - 2$
Let $f\left( x \right) = {x^3} + 3{x^2} + 5x + 6$
If $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$ , then $f\left( x \right)$ will be zero at $x = - 2$ .
Now we can find $f\left( { - 2} \right)$ . For that we can substitute $x = - 2$ in all the places of x.
 $ \Rightarrow f\left( { - 2} \right) = {\left( { - 2} \right)^3} + 3{\left( { - 2} \right)^2} + 5\left( { - 2} \right) + 6$
On further simplification, we get,
 $ \Rightarrow f\left( { - 2} \right) = - 8 + 12 - 10 + 6$
 $ \Rightarrow f\left( { - 2} \right) = 0$
As $x = - 2$ is the root of $x + 2$ and $f\left( x \right) = {x^3} + 3{x^2} + 5x + 6$ is equal to zero at $x = - 2$ , we can say that $x + 2$ is a factor of ${x^3} + 3{x^2} + 5x + 6$ .
So, the given expression is a factor of the polynomial.