Chapter 4.6, Problem 47WE

a.

To determine

To Find:

To find the GCF of the polynomial $t^3-2t^2-3t;t^3+5t^2+4t$ .

Answer to Problem 47WE

The GCF of the polynomial $t^3-2t^2-3t;t^3+5t^2+4t$ is $t\left(t+1\right)$ .

Explanation of Solution

Given information:

The given polynomial is $t^3-2t^2-3t;t^3+5t^2+4t$ .

To Factorize $t^3-2t^2-3t$ take the common term out,

  $=t\left(t^2-2t-3\right)$

Now factorize: $\left(t^2-2t-3\right)$ ,

  $=\left(t+1\right)\left(t-3\right)$

  $t\left(t^2-2t-3\right)\Rightarrow t\left(t+1\right)\left(t-3\right)$

To Factorize $t^3+5t^2+4t$ take the common term out,

  $=t\left(t^2+5t+4\right)$

Now further factorize: $\left(t^2+5t+4\right)$ ,

  $\Rightarrow t\left(t+1\right)\left(t+4\right)$

The greatest common divisor of $t\left(t+1\right)\left(t-3\right)$ and $t\left(t+1\right)\left(t+4\right)$ are $t\left(t+1\right)$ .

Therefore, the GCF of the polynomial $t^3-2t^2-3t;t^3+5t^2+4t$ is $t\left(t+1\right)$ .

b.

To determine

To Find:

To find the LCM of the polynomial $t^3-2t^2-3t;t^3+5t^2+4t$ .

Answer to Problem 47WE

The LCM of the polynomial $t^3-2t^2-3t;t^3+5t^2+4t$ is $t\left(t+1\right)\left(t-3\right)\left(t+4\right)$ .

Explanation of Solution

Given information:

The given polynomial is $t^3-2t^2-3t;t^3+5t^2+4t$ .

To Factorize $t^3-2t^2-3t$ take the common term out,

  $=t\left(t^2-2t-3\right)$

Now factorize: $\left(t^2-2t-3\right)$ ,

  $=\left(t+1\right)\left(t-3\right)$

  $t\left(t^2-2t-3\right)\Rightarrow t\left(t+1\right)\left(t-3\right)$

To Factorize $t^3+5t^2+4t$ take the common term out,

  $=t\left(t^2+5t+4\right)$

Now further factorize: $\left(t^2+5t+4\right)$ ,

  $\Rightarrow t\left(t+1\right)\left(t+4\right)$

Compute an expression of factors that appears either in $t\left(t+1\right)\left(t-3\right)$ or $t\left(t+1\right)\left(t+4\right)$ .

Therefore, the LCM of the polynomial $t^3-2t^2-3t;t^3+5t^2+4t$ is $t\left(t+1\right)\left(t-3\right)\left(t+4\right)$ .