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]
