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The expression [tex]$9x^2 - ax - 76$[/tex], where [tex]$a$[/tex] is a constant, can be rewritten as [tex]$(bx + c)(x + d)$[/tex], where [tex][tex]$b$[/tex][/tex], [tex]$c$[/tex], and [tex]$d$[/tex] are integer constants. Which of the following must be an integer?

A) [tex]$\frac{a}{b}$[/tex]

B) [tex][tex]$\frac{a}{c}$[/tex][/tex]

C) [tex]$\frac{76}{b}$[/tex]

D) [tex]$\frac{76}{d}$[/tex]

Sagot :

GinnyAnswer
To solve the problem of determining which of the given expressions must be an integer, we need to factor the quadratic expression [tex]\(9x^2 - ax - 76\)[/tex] into a product of linear polynomials [tex]\((bx + c)(x + d)\)[/tex].

First, let’s expand [tex]\((bx + c)(x + d)\)[/tex]:
[tex]\[ (bx + c)(x + d) = bx \cdot x + bx \cdot d + c \cdot x + c \cdot d = bx^2 + (bd + c)x + cd \][/tex]

We need this to be equal to [tex]\(9x^2 - ax - 76\)[/tex]. Comparing coefficients from the expanded form, we get:
[tex]\[ bx^2 + (bd + c)x + cd = 9x^2 - ax - 76 \][/tex]

From this, we can extract the following system of equations by equating coefficients:
1. [tex]\(b = 9\)[/tex]
2. [tex]\(cd = -76\)[/tex]
3. [tex]\(bd + c = -a\)[/tex]

Let's analyze each option given:
- Option A: [tex]\(\frac{a}{b}\)[/tex]
- Here, [tex]\( b = 9 \)[/tex]. We need [tex]\(a\)[/tex] to be an integer such that [tex]\( \frac{a}{9} \)[/tex] is also an integer.

- Option B: [tex]\(\frac{a}{c}\)[/tex]
- Here, we would need [tex]\(a\)[/tex] to be an integer such that [tex]\( \frac{a}{c} \)[/tex] is an integer, which depends on the specific values of [tex]\(c\)[/tex].

- Option C: [tex]\(\frac{76}{b}\)[/tex]
- The value of [tex]\(b\)[/tex] is [tex]\(9\)[/tex]. Hence, [tex]\(\frac{76}{9}\)[/tex] is not an integer, since 76 is not divisible by 9. Therefore, this option cannot be an integer.

- Option D: [tex]\(\frac{76}{d}\)[/tex]
- Here, [tex]\(d\)[/tex] could be any factor of 76. Specifically, if we consider the products that give [tex]\(cd = -76\)[/tex], we see that [tex]\(c\)[/tex] and [tex]\(d\)[/tex] are pairs like [tex]\( (1, -76), (2, -38), (4, -19), etc.\)[/tex]. We immediately see that regardless of the pair chosen, [tex]\(d\)[/tex] must be an integer factor of 76, so this expression is always an integer.

Let us verify:
- The factors of 76 are [tex]\( \pm1, \pm2, \pm4, \pm19, \pm38, \pm76 \)[/tex].
Therefore, [tex]\(d\)[/tex] is always an integer factor of 76, which means [tex]\(\frac{76}{d}\)[/tex] must always be an integer.

Thus, the correct answer is:
D) [tex]\(\frac{76}{d}\)[/tex]
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