Chapter 1.3, Problem 23E

To determine

To calculate: The simplified value of the expression $\left(3t-2\right)\left(7t-4\right)$ .

Answer to Problem 23E

The value of the expression $\left(3t-2\right)\left(7t-4\right)$ is $21t^2-26t+8$ .

Explanation of Solution

Given information:

The expression $\left(3t-2\right)\left(7t-4\right)$ .

Formula used:

The FOIL method to multiply two polynomial expression is the sum of the products of its First terms, Outer terms, Inner terms and the Last terms which can be mathematically expressed as, $\left(\text{a+b}\right)\left(\text{c}+\text{d}\right)=\text{ac}+\text{ad+bc+bd}$ .

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

Calculation:

Consider the given expression $\left(3t-2\right)\left(7t-4\right)$ .

Recall the FOIL method to multiply two polynomial expression is the sum of the products of its First terms, Outer terms, Inner terms and the Last terms which can be mathematically expressed as, $\left(\text{a+b}\right)\left(\text{c}+\text{d}\right)=\text{ac}+\text{ad+bc+bd}$ .

Apply it,

  $\begin{array}{c}\left(3t-2\right)\left(7t-4\right)=\left(3t\cdot 7t\right)+\left(3t\cdot \left(-4\right)\right)+\left(\left(-2\right)\cdot 7t\right)+\left(\left(-2\right)\cdot \left(-4\right)\right)\\ =21t^2-12t-14t+8\end{array}$

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,

  $\begin{array}{c}\left(3t-2\right)\left(7t-4\right)=21t^2-12t-14t+8\\ =21t^2+\left(-12t-14t\right)+8\\ =21t^2-26t+8\end{array}$

Thus, the value of the expression $\left(3t-2\right)\left(7t-4\right)$ is $21t^2-26t+8$ .