Chapter 1.3, Problem 21E

To determine

To calculate: The value of the expression $2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)$ .

Answer to Problem 21E

The value of the expression $2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)$ is $-t^4+t^3-t^2-10t+5$ .

Explanation of Solution

Given information:

The expression $2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)$ .

Formula used:

To multiply the polynomials or algebraic expressions apply the distributive property which is expressed as $\text{a}\left(\text{b}+\text{c}\right)=\text{ab}+\text{ac}$ .

To add or subtract two polynomials, combine all the like terms together and then perform the operation.

Calculation:

Consider the given expression $2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)$ .

Recall that to multiply the polynomials or algebraic expressions apply the distributive property which is expressed as $\text{a}\left(\text{b}+\text{c}\right)=\text{ab}+\text{ac}$ .

Apply it,

  $2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)=4-10t+t^3-t^2-t^4+1$

Recall that to add or subtract two polynomials, combine all the like terms together and then perform the operation.

Apply it,

So, the value of the expression will be calculated as,

  $\begin{array}{c}2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)=4-10t+t^3-t^2-t^4+1\\ =-t^4+t^3-t^2-10t+\left(4+1\right)\\ =-t^4+t^3-t^2-10t+5\end{array}$

Thus, the value of the expression $2\left(2-5t\right)+t^2\left(t-1\right)-\left(t^4-1\right)$ is $-t^4+t^3-t^2-10t+5$ .