Chapter 8.7, Problem 5CGP

To determine

To factor: The given polynomial.

Answer to Problem 5CGP

The factored polynomial is $2m^3+m^2-50m-25=\left(m-5\right)\left(m-5\right)\left(2m-1\right)$ .

Explanation of Solution

Given information:

The given polynomial is $2m^3+m^2-50m-25$ .

Concept used:

The cubic polynomial is factorized by finding a number for which the given polynomial is zero.

Calculations:

The given polynomial E (say) is $E\left(m\right)=2m^3+m^2-50m-25$ .

We factor the constant term $-25=5\times 5$ . Now, we find from -5 or +5 for which the given polynomial is zero. We consider $m=5$

Now, $E\left(5\right)=2{\left(5\right)}^3+{\left(5\right)}^2-50\left(5\right)-25$

  $\Rightarrow E\left(5\right)=250+25-250-25=\cancel{250}\cancel{+25}\cancel{-250}\cancel{-25}=0$............(1)

Since, for $m=5$ , the polynomial is zero, therefore, $\left(m-5\right)$ is a factor of the polynomial E .

Now, since given polynomial is cubic so there are 3 liner factor of it or if we divide it by $\left(m-5\right)$ we get a quotient, a quadratic polynomial.

Let that quadratic polynomial is $a{m}^2+bm+c$ , then

  $E\left(m\right)=2m^3+m^2-50m-25=\left(m-5\right)\left(a{m}^2+bm+c\right)$

  $\Rightarrow 2m^3+m^2-50m-25=\left(a{m}^3+b{m}^2+cm-5a{m}^2-5bm-5c\right)$

  $\Rightarrow 2m^3+m^2-50m-25=a{m}^3+\left(b-5a\right)m^2+\left(c-5b\right)m-5c$......................(2)

Equating coefficients of like power of m , we have $a=2$ , $\left(b-5a\right)=1$ , $\left(c-5b\right)=-50$ , and $-5c=-25$ .

Thus, $a=2$

  $b=11$ , and $c=5$ . Therefore,

  $E\left(m\right)=2m^3+m^2-50m-25=\left(m-5\right)\left(2m^2-11m+5\right)$ ...........(3)

To factor $2m^2-11m+5$ , we find two factors of $2\times 5=10$ such that their sum is 11. These are 10 and 1 as $2\times 5=10\times 1$ , $10+1=11$ .

  $2m^2-11m+5=2m^2-10m-m+5$

  $\Rightarrow 2m^2-11m+5=2m\left(m-5\right)-1\left(m-5\right)$

  $\Rightarrow 2m^2-11m+5=\left(m-5\right)\left(2m-1\right)$............(4)

From results (3) and (4)

  $E\left(m\right)=2m^3+m^2-50m-25=\left(m-5\right)\left(m-5\right)\left(2m-1\right)$

Thus, the factored polynomial is $2m^3+m^2-50m-25=\left(m-5\right)\left(m-5\right)\left(2m-1\right)$ .

Conclusion:

The factored polynomial is $2m^3+m^2-50m-25=\left(m-5\right)\left(m-5\right)\left(2m-1\right)$ .