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VI. Write a polynomial in each [tex]$\square$[/tex] to complete each statement.

34. [tex]$7p^2 - 7p = 7p \cdot \square$[/tex]

35. [tex]$6a^2b + 24a^3 = 6a^2 \cdot \square$[/tex]

36. [tex]$35x^5y^2 + 21x^4y + 14x^3y^2 = 7x^3y \cdot \square$[/tex]

37. [tex]$10b^3 - 80b^5 - 5b^6 + 5b^7 = 5b^3 \cdot \square$[/tex]

38. [tex]$17x^5 - 51x^4 - 34x = \square \cdot (x^4 - 3x^3 - 2)$[/tex]

39. [tex]$\frac{2}{5}x - \frac{1}{2} = \frac{1}{10} \cdot \square$[/tex]

Sagot :

GinnyAnsweridn
Certainly! Let's tackle each problem step-by-step.

### Problem 34
[tex]\[ 7p^2 - 7p = 7p \square \][/tex]

First, let's rewrite the left-hand side:
[tex]\[ 7p^2 - 7p = 7(p^2 - p) \][/tex]

Therefore, the [tex]\(\square\)[/tex] should be:
[tex]\[ \square = p^2 - p \][/tex]

So the complete statement is:
[tex]\[ 7p^2 - 7p = 7p(p - 1) \][/tex]

### Problem 35
[tex]\[ 6a^2b + 24a^3 = 6a^2 \square \][/tex]

Factor out the greatest common factor (GCF) from the left side:
[tex]\[ 6a^2b + 24a^3 = 6a^2(b + 4a) \][/tex]

Therefore, the [tex]\(\square\)[/tex] should be:
[tex]\[ \square = b + 4a \][/tex]

So the complete statement is:
[tex]\[ 6a^2b + 24a^3 = 6a^2(b + 4a) \][/tex]

### Problem 36
[tex]\[ 35x^5y^2 + 21x^4y + 14x^3y^2 = 7x^3y \square \][/tex]

Factor out the GCF:
[tex]\[ 35x^5y^2 + 21x^4y + 14x^3y^2 = 7x^3y(5x^2y + 3x + 2y) \][/tex]

Therefore, the [tex]\(\square\)[/tex] should be:
[tex]\[ \square = 5x^2y + 3x + 2y \][/tex]

So the complete statement is:
[tex]\[ 35x^5y^2 + 21x^4y + 14x^3y^2 = 7x^3y(5x^2y + 3x + 2y) \][/tex]

### Problem 37
[tex]\[ 10b^3 - 80b^5 - 5b^6 + 5b^7 = 5b^3 \square \][/tex]

Factor out the GCF:
[tex]\[ 10b^3 - 80b^5 - 5b^6 + 5b^7 = 5b^3(2 - 16b^2 - b^3 + b^4) \][/tex]

Therefore, the [tex]\(\square\)[/tex] should be:
[tex]\[ \square = 2 - 16b^2 - b^3 + b^4 \][/tex]

So the complete statement is:
[tex]\[ 10b^3 - 80b^5 - 5b^6 + 5b^7 = 5b^3(2 - 16b^2 - b^3 + b^4) \][/tex]

### Problem 38
[tex]\[ 17x^5 - 51x^4 - 34x = \square (x^4 - 3x^3 - 2) \][/tex]

Notice that [tex]\( x = 0 \)[/tex] is a common root. Let’s factor out [tex]\( x \)[/tex]:
[tex]\[ 17x^5 - 51x^4 - 34x = x(17x^4 - 51x^3 - 34) \][/tex]

So [tex]\(\square\)[/tex] should be:
[tex]\[ \square = 17x \][/tex]

Therefore, the complete statement is:
[tex]\[ 17x^5 - 51x^4 - 34x = 17x(x^4 - 3x^3 - 2) \][/tex]

### Problem 39
[tex]\[ \frac{2}{5}x - \frac{1}{2} = \frac{1}{10} \square \][/tex]

Let's isolate [tex]\(\frac{1}{10} \square\)[/tex]:
[tex]\[ \frac{2}{5}x - \frac{1}{2} = \frac{1}{10} \square \][/tex]
Multiply both sides by 10 to clear the fraction:
[tex]\[ 4x - 5 = \square \][/tex]

So [tex]\(\square\)[/tex] should be:
[tex]\[ \square = 4x - 5 \][/tex]

Therefore, the complete statement is:
[tex]\[ \frac{2}{5}x - \frac{1}{2} = \frac{1}{10}(4x - 5) \][/tex]

### Summary of Answers
34. [tex]\( \square = p(p - 1) \)[/tex]
35. [tex]\( \square = b + 4a \)[/tex]
36. [tex]\( \square = 5x^2y + 3x + 2y \)[/tex]
37. [tex]\( \square = 2 - 16b^2 - b^3 + b^4 \)[/tex]
38. [tex]\( \square = 17x \)[/tex]
39. [tex]\( \square = 4x - 5 \)[/tex]
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