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Sagot :
To determine which of the given options must be an integer, let’s factorize the quadratic expression [tex]\( 4x^2 + bx - 45 \)[/tex] into the form [tex]\( (hx + k)(x + j) \)[/tex], where [tex]\( h \)[/tex], [tex]\( k \)[/tex], and [tex]\( j \)[/tex] are integer constants.
The expanded form of the factorized expression is:
[tex]\[ (hx + k)(x + j) = hx^2 + h j x + k x + k j \][/tex]
Upon expanding, we combine like terms:
[tex]\[ hx^2 + (hj + k)x + kj \][/tex]
We need this to match the form [tex]\( 4x^2 + bx - 45 \)[/tex]. Therefore:
[tex]\[ hx^2 + (hj + k)x + kj = 4x^2 + bx - 45 \][/tex]
From this comparison, we can identify the coefficients:
[tex]\[ h = 4, \quad hj + k = b, \quad kj = -45 \][/tex]
Given [tex]\( h = 4 \)[/tex], we substitute [tex]\( 4 \)[/tex] into the equations:
[tex]\[ 4j + k = b \tag{1} \][/tex]
[tex]\[ k j = -45 \tag{2} \][/tex]
From Equation (2), since [tex]\( k \)[/tex] and [tex]\( j \)[/tex] are integers, we look for integer pairs [tex]\((k, j)\)[/tex] such that their product is [tex]\(-45\)[/tex]. Here are the possible integer pairs [tex]\((k, j)\)[/tex]:
[tex]\[ (k, j) = (1, -45), (-1, 45), (3, -15), (-3, 15), (5, -9), (-5, 9), (9, -5), (-9, 5), (15, -3), (-15, 3), (45, -1), (-45, 1) \][/tex]
For each pair, we need to check if [tex]\( b = 4j + k \)[/tex] remains an integer.
Now let's examine the four options given in the question to determine which one must be an integer:
### Option A: [tex]\(\frac{b}{h}\)[/tex]
Substituting [tex]\( h = 4 \)[/tex]:
[tex]\[ \frac{b}{4} = \frac{4j + k}{4} \][/tex]
This is not necessarily an integer because [tex]\( \frac{k}{4} \)[/tex] may not be an integer (for example, if [tex]\( k = 3 \)[/tex]).
### Option B: [tex]\(\frac{b}{k}\)[/tex]
Substituting [tex]\( b = 4j + k \)[/tex]:
[tex]\[ \frac{b}{k} = \frac{4j + k}{k} = 4\frac{j}{k} + 1 \][/tex]
This is not necessarily an integer because [tex]\( 4\frac{j}{k} \)[/tex] may not be an integer (for example, if [tex]\( j = 5 \)[/tex] and [tex]\( k = 9 \)[/tex]).
### Option C: [tex]\(\frac{45}{h}\)[/tex]
Substituting [tex]\( h = 4 \)[/tex]:
[tex]\[ \frac{45}{4} \][/tex]
This is not an integer.
### Option D: [tex]\(\frac{45}{k}\)[/tex]
Since [tex]\( kj = -45 \)[/tex] and both [tex]\( k \)[/tex] and [tex]\( j \)[/tex] are integers, [tex]\( k \)[/tex] must be a divisor of 45. Therefore, each possible [tex]\( k \)[/tex] (from the pairs given) is a divisor of 45:
[tex]\[ k \in \{\pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45\} \][/tex]
Thus, [tex]\( \frac{45}{k} \)[/tex] must be an integer.
Therefore, the option that must be an integer is:
[tex]\[ \boxed{\text{D}} \][/tex]
The expanded form of the factorized expression is:
[tex]\[ (hx + k)(x + j) = hx^2 + h j x + k x + k j \][/tex]
Upon expanding, we combine like terms:
[tex]\[ hx^2 + (hj + k)x + kj \][/tex]
We need this to match the form [tex]\( 4x^2 + bx - 45 \)[/tex]. Therefore:
[tex]\[ hx^2 + (hj + k)x + kj = 4x^2 + bx - 45 \][/tex]
From this comparison, we can identify the coefficients:
[tex]\[ h = 4, \quad hj + k = b, \quad kj = -45 \][/tex]
Given [tex]\( h = 4 \)[/tex], we substitute [tex]\( 4 \)[/tex] into the equations:
[tex]\[ 4j + k = b \tag{1} \][/tex]
[tex]\[ k j = -45 \tag{2} \][/tex]
From Equation (2), since [tex]\( k \)[/tex] and [tex]\( j \)[/tex] are integers, we look for integer pairs [tex]\((k, j)\)[/tex] such that their product is [tex]\(-45\)[/tex]. Here are the possible integer pairs [tex]\((k, j)\)[/tex]:
[tex]\[ (k, j) = (1, -45), (-1, 45), (3, -15), (-3, 15), (5, -9), (-5, 9), (9, -5), (-9, 5), (15, -3), (-15, 3), (45, -1), (-45, 1) \][/tex]
For each pair, we need to check if [tex]\( b = 4j + k \)[/tex] remains an integer.
Now let's examine the four options given in the question to determine which one must be an integer:
### Option A: [tex]\(\frac{b}{h}\)[/tex]
Substituting [tex]\( h = 4 \)[/tex]:
[tex]\[ \frac{b}{4} = \frac{4j + k}{4} \][/tex]
This is not necessarily an integer because [tex]\( \frac{k}{4} \)[/tex] may not be an integer (for example, if [tex]\( k = 3 \)[/tex]).
### Option B: [tex]\(\frac{b}{k}\)[/tex]
Substituting [tex]\( b = 4j + k \)[/tex]:
[tex]\[ \frac{b}{k} = \frac{4j + k}{k} = 4\frac{j}{k} + 1 \][/tex]
This is not necessarily an integer because [tex]\( 4\frac{j}{k} \)[/tex] may not be an integer (for example, if [tex]\( j = 5 \)[/tex] and [tex]\( k = 9 \)[/tex]).
### Option C: [tex]\(\frac{45}{h}\)[/tex]
Substituting [tex]\( h = 4 \)[/tex]:
[tex]\[ \frac{45}{4} \][/tex]
This is not an integer.
### Option D: [tex]\(\frac{45}{k}\)[/tex]
Since [tex]\( kj = -45 \)[/tex] and both [tex]\( k \)[/tex] and [tex]\( j \)[/tex] are integers, [tex]\( k \)[/tex] must be a divisor of 45. Therefore, each possible [tex]\( k \)[/tex] (from the pairs given) is a divisor of 45:
[tex]\[ k \in \{\pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45\} \][/tex]
Thus, [tex]\( \frac{45}{k} \)[/tex] must be an integer.
Therefore, the option that must be an integer is:
[tex]\[ \boxed{\text{D}} \][/tex]
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