To identify the error in the given verbal description of the algebraic expression [tex]\(\frac{7 x^2(2 x - 3)}{3 x^3 + 4}\)[/tex], let's break down both the verbal description and the mathematical expression step by step.
### Analyzing the Mathematical Expression:
The mathematical expression is:
[tex]\[
\frac{7 x^2(2 x - 3)}{3 x^3 + 4}
\][/tex]
### Breaking Down Each Component:
1. Numerator: [tex]\(7 x^2(2 x - 3)\)[/tex]
- [tex]\(7 x^2\)[/tex] means "seven times the square of [tex]\(x\)[/tex]".
- [tex]\((2 x - 3)\)[/tex] is "the difference of twice [tex]\(x\)[/tex] and three".
2. Denominator: [tex]\(3 x^3 + 4\)[/tex]
- [tex]\(3 x^3\)[/tex] means "three times the cube of [tex]\(x\)[/tex]".
- The entire denominator is the expression [tex]\(3 x^3 + 4\)[/tex], which means "the sum of three times the cube of [tex]\(x\)[/tex] and four".
### Verifying the Verbal Description:
The verbal description given is:
"the product of seven times the square of [tex]\(x\)[/tex] and the difference of twice [tex]\(x\)[/tex] and three divided by three times the sum of the cube of [tex]\(x\)[/tex] and four"
### Identifying the Error:
1. Numerator Description:
- "the product of seven times the square of [tex]\(x\)[/tex]"
- "and the difference of twice [tex]\(x\)[/tex] and three"
This part correctly describes [tex]\(7 x^2 (2 x - 3)\)[/tex].
2. Denominator Description:
- "divided by three times the sum of the cube of [tex]\(x\)[/tex] and four"
Let's focus on this part:
- "three times the sum of the cube of [tex]\(x\)[/tex] and four" implies [tex]\(3 \cdot (x^3 + 4)\)[/tex], which suggests that the entire expression [tex]\(x^3 + 4\)[/tex] is multiplied by 3.
- However, the denominator [tex]\(3 x^3 + 4\)[/tex] means [tex]\(3 x^3\)[/tex] (three times the cube of [tex]\(x\)[/tex]) added to 4, not the sum inside a product.
### Conclusion:
The error in the verbal description lies in the phrase:
"three times the sum of the cube of [tex]\(x\)[/tex] and four"
This phrase incorrectly suggests that the entire sum [tex]\(x^3 + 4\)[/tex] is multiplied by 3.
The correct phrase should be:
"the sum of three times the cube of [tex]\(x\)[/tex] and four"
### Final Answer:
The phrase containing an error is:
```
three times the sum of the cube of [tex]\(x\)[/tex] and four
```
