Question

How do you factor by grouping $3{{p}^{2}}-2p-5$?

Answer 1

Hint: The given polynomial, $3{{p}^{2}}-2p-5$, is a quadratic polynomial. So we can use the middle term splitting method to factor it. Using the middle term splitting method, we have to split the middle term of the given polynomial, which is $-2p$, into two terms such that their product is equal to the product of the first term, $3{{p}^{2}}$, and the second term, $-5$, that is, equal to $-15{{p}^{2}}$. So we can split the middle term as $-2p=3p-5p$ to write the given polynomial as $3{{p}^{2}}+3p-5p-5$. Then we have to group the first two and the last two terms as $\left( 3{{p}^{2}}+3p \right)+\left( -5p-5 \right)$. From these groups we can take $3p$ and $-5$ common to finally factor the polynomial.

Complete step by step solution:
The given polynomial is in the terms of the variable $p$. So it can be written as
\[\Rightarrow f\left( p \right)=3{{p}^{2}}-2p-5\]
To use the factor by grouping method in a polynomial, we need an even number of terms in the polynomial. We have three terms in the above polynomial, which is odd. So we have to split the middle term of the above polynomial using the middle term splitting method to get
$\Rightarrow f\left( p \right)=3{{p}^{2}}+3p-5p-5$
Now, we group the first two and the last two terms to get
$\Rightarrow f\left( p \right)=\left( 3{{p}^{2}}+3p \right)+\left( -5p-5 \right)$
We can take $3p$ and $-5$ common from the two groups to get
$\Rightarrow f\left( p \right)=3p\left( p+1 \right)-5\left( p+1 \right)$
Finally, we take $\left( p+1 \right)$ common to get
$\Rightarrow f\left( p \right)=\left( p+1 \right)\left( 3p-5 \right)$
Hence, the given polynomial is factored by grouping as $\left( p+1 \right)\left( 3p-5 \right)$.

Note:
For checking whether we have factored the given polynomial or not, we can equate each of the obtained factors to zero to obtain the zeroes of the polynomial. Then on substituting those zeroes into the given polynomial, we must obtain zero. Then only by the factor theorem we can say that the given polynomial is factored correctly.