Question

How do you factor by grouping ${{x}^{2}}+7x+5x+35$?

Answer 1

Hint: In the given polynomial ${{x}^{2}}+7x+5x+35$, first we need to check whether the product of the middle two terms is equal to the product of the first and last terms or not. The middle terms are $7x$ and $5x$, which means that their product is equal to $35{{x}^{2}}$. Also, the first and the last terms are ${{x}^{2}}$ and $35$, which makes the product equal to $35{{x}^{2}}$. Since the two products are equal, we can use the factor by grouping method. For this, we need to combine the first two and the last two terms so that we have two groups of the terms as $\left( {{x}^{2}}+7x \right)+\left( 5x+35 \right)$. Then on taking $x$ and $5$ common from the pairs, the given polynomial will be factored.

Complete step by step solution:
The polynomial given in the above question is
$\Rightarrow p\left( x \right)={{x}^{2}}+7x+5x+35$
We can use the factor by grouping method only when the product of the middle two terms is equal to the product of the remaining two terms. The middle terms are $7x$ and $5x$, so that their product is equal to $35{{x}^{2}}$. We can easily observe that this product is equal to the product of the remaining terms ${{x}^{2}}$ and $35$. So we can use the factor by grouping method. For this we combine the first two and the last two terms to get
$\Rightarrow p\left( x \right)=\left( {{x}^{2}}+7x \right)+\left( 5x+35 \right)$
Now, we take the common factors $x$ and $5$ outside from the first and the second groups to get
$\Rightarrow p\left( x \right)=x\left( x+7 \right)+5\left( x+7 \right)$
Finally, we take the factor $\left( x+7 \right)$ common to get
$\Rightarrow p\left( x \right)=\left( x+7 \right)\left( x+5 \right)$
Hence, the given polynomial is factored by grouping.

Note:
It is important to check for the two products of the middle two and the remaining two terms whether they are equal or not. This is because only then we will be able to use the factor by grouping method to factor the given polynomial. If the two products are not equal, then we have to combine the two middle terms and again split it according to the middle term splitting method.