Chapter 12, Problem 17CLR

To determine

To Evaluate: The simplified value of the expression $\left(4u^2-2u+1\right)\left(2u+1\right)$ .

Answer to Problem 17CLR

The value of the expression $\left(4u^2-2u+1\right)\left(2u+1\right)$ is $6u^3+1$ .

Explanation of Solution

Given information:

The expression is $\left(4u^2-2u+1\right)\left(2u+1\right)$ .

Formula used:

When an algebraic expression multiply by group of algebraic terms connected with addition or subtraction is equal to multiplication of expression with the terms separately. That is $a\left(b\pm c\right)=\left(a\times b\right)\pm \left(a\times c\right)$

When an algebraic term is multiply by another algebraic term power of same variable added and the numerical coefficients multiply together.

We can add terms of an algebraic expression which is like.

Calculation:

Consider the expression is $\left(4u^2-2u+1\right)\left(2u+1\right)$ .

Recall that when an algebraic expression multiply by group of algebraic terms connected with addition or subtraction is equal to multiplication of expression with the terms separately. That is $a\left(b\pm c\right)=\left(a\times b\right)\pm \left(a\times c\right)$

Use it,

  $\begin{array}{l}\left(4u^2-2u+1\right)\left(2u+1\right)=\left(\left(4u^2-2u+1\right)\times 2u\right)+\left(\left(4u^2-2u+1\right)\times 1\right)\\ =\left(\left(4u^2\times 2u\right)-\left(2u\times 2u\right)+\left(1\times 2u\right)\right)+\left(4u^2-2u+1\right)\end{array}$

Again recall that when a algebraic term is multiply by another algebraic term power of same variable added and the numerical coefficients multiply together.

Apply it,

  $\begin{array}{c}\left(\left(4u^2\times 2u\right)-\left(2u\times 2u\right)+\left(1\times 2u\right)\right)+\left(4u^2-2u+1\right)=\left(6u^3-4u^2+2u\right)+\left(4u^2-2u+1\right)\\ =6u^3-4u^2+2u+4u^2-2u+1\end{array}$

Also we know that we can add terms of an algebraic expression which is like.

Further simplify,

  $\begin{array}{c}6u^3-4u^2+2u+4u^2-2u+1=6u^3+\left(-4u^2+4u^2\right)+\left(2u-2u\right)+1\\ =6u^3+0+0+1\\ =6u^3+1\end{array}$

Thus, the value of the expression $\left(4u^2-2u+1\right)\left(2u+1\right)$ is $6u^3+1$ .