Chapter 8.2, Problem 54PFA

A.

To determine

Tofind: If the statement for the given expressions are true or not.

Answer to Problem 54PFA

False

Explanation of Solution

Given:

The expressions:

I. $2.5x^2\left(2x^3-10x^2+x\right)$

II. $6y{z}^3\left(9y^2-4z\right)$

III. $v^{2}w\left(w^3+3v^2-180v^5\right)$

Calculation:

Find the value of the expression I when $x=1$ .

  $\begin{array}{c}2.5x^2\left(2x^3-10x^2+x\right)=2.5{\left(1\right)}^2\left(2{\left(1\right)}^3-10{\left(1\right)}^2+1\right)\\ =2.5\left(2-10+1\right)\\ =2.5\left(-7\right)\\ =-17.5\end{array}$

The value of the expression I is not greater than 0 when $x=1$ . So, the statement is false.

B.

To determine

To find: If the statement for the given expressions are true or not.

Answer to Problem 54PFA

True

Explanation of Solution

Given:

The expressions:

I. $2.5x^2\left(2x^3-10x^2+x\right)$

II. $6y{z}^3\left(9y^2-4z\right)$

III. $v^{2}w\left(w^3+3v^2-180v^5\right)$

Statement: Two of the expressions have degree greater than 5.

Calculation:

Observe that the expression I has degree 5, expression II has degree 6 and expression III has degree 8.

So, the expressions II and III have degree greater than 5.

Therefore, the statement is true.

C.

To determine

To find: If the statement for the given expressions are true or not.

Answer to Problem 54PFA

False

Explanation of Solution

Given:

The expressions:

I. $2.5x^2\left(2x^3-10x^2+x\right)$

II. $6y{z}^3\left(9y^2-4z\right)$

III. $v^{2}w\left(w^3+3v^2-180v^5\right)$

Statement: the value of the expression III is negative when $v=-2$ and $w=2$ .

Calculation:

Substitute $v=-2$ and $w=2$ in the expression III and simplify to check the value of the expression.

  $\begin{array}{c}{v}^{2}w\left(w^3+3v^2-180v^5\right)={\left(-2\right)}^2\cdot 2\left[2^3+3{\left(-2\right)}^2-180{\left(-2\right)}^5\right]\\ =4\cdot 2\left[8+3\cdot 4-180\left(-32\right)\right]\\ =8\left(8+12+\text{5760}\right)\\ =8\cdot 5780\\ =\text{46080}\end{array}$

Since the value of the expression is positive, the given statement is not true.

D.

To determine

To find: If the statement for the given expressions are true or not.

Answer to Problem 54PFA

True

Explanation of Solution

Given:

The expressions:

I. $2.5x^2\left(2x^3-10x^2+x\right)$

II. $6y{z}^3\left(9y^2-4z\right)$

III. $v^{2}w\left(w^3+3v^2-180v^5\right)$

Statement: the expression III has the greatest degree.

Calculation:

Since the expression I has degree 5, expression II has degree 6 and expression III has degree 8, the expression III has greatest degree.

Therefore, the statement is true.

E.

To determine

To find: If the statement for the given expressions are true or not.

Answer to Problem 54PFA

True

Explanation of Solution

Given:

The expressions:

I. $2.5x^2\left(2x^3-10x^2+x\right)$

II. $6y{z}^3\left(9y^2-4z\right)$

III. $v^{2}w\left(w^3+3v^2-180v^5\right)$

Statement: The simplified form of the expression II is $54y^3{z}^3-24y{z}^4$ .

Calculation:

Simplify the expression II, to check if the statement is true or not.

  $\begin{array}{c}6y{z}^3\left(9y^2-4z\right)=6y{z}^3\cdot 9y^2-6y{z}^3\cdot 4z\left(\text{Distribute }6y{z}^3\text{ over parenthesis}\right)\\ =54y^{1+2}{z}^3-24y{z}^{3+1}\left(\text{Use exponent rule:}\text{}{a}^m\cdot a^n=a^{m+n}\right)\\ =54y^3{z}^3-24y{z}^4\end{array}$

So, the simplified form of the expression II is $54y^3{z}^3-24y{z}^4$ .

Therefore, the statement is true.

F.

To determine

To find: If the statement for the given expressions are true or not.

Answer to Problem 54PFA

True

Explanation of Solution

Given:

The expressions:

I. $2.5x^2\left(2x^3-10x^2+x\right)$

II. $6y{z}^3\left(9y^2-4z\right)$

III. $v^{2}w\left(w^3+3v^2-180v^5\right)$

Statement: The expression I has a degree 5.

Calculation:

Distribute $2.5x^2$ over the parenthesis to find the simplified form of the expression I can be written as $5x^5-25x^4+2.5x^3$ .

Thus, the degree of the expression I is 5.

So, the given statement is true.