Answer:
[tex]n = 241[/tex]
Step-by-step explanation:
Given
[tex]5x^2 + nx + 48[/tex]
Required
Determine the highest value of n
From the given equation, 5 is a prime number;
So, the factors of x² is 5x and x or -5x and -x
Since [tex]5x^2 + nx + 48[/tex] has all shades of positive terms, we'll make use of 5x and x
The factorized expression can then be:
[tex](5x + a)(x + b)[/tex]
Open the brackets
[tex]5x^2 + ax + 5bx + ab[/tex]
Equate this to the given expression
[tex]5x^2 + ax + 5bx + ab = 5x^2 + nx + 48[/tex]
[tex]5x^2 + (a + 5b)x + ab = 5x^2 + nx + 48[/tex]
By direct comparison;
[tex]5x^2 = 5x^2[/tex]
[tex](a + 5b)x = nx[/tex]
[tex]a + 5b = n[/tex] ---- (1)
[tex]ab = 48[/tex] --- (2)
From (2) above, the possible values of a and b are:
[tex]a = 1, b = 48[/tex]
[tex]a = 2, b = 24[/tex]
[tex]a = 3, b = 16[/tex]
[tex]a = 4, c = 12[/tex]
[tex]a = 6, b = 8[/tex]
[tex]a = 8, b = 6[/tex]
[tex]a = 12, b = 4[/tex]
[tex]a = 16, b = 3[/tex]
[tex]a = 24, b = 2[/tex]
[tex]a = 48, b = 1[/tex]
Of all these values; the value of a and b that gives the highest value of n is;
[tex]a = 1, b = 48[/tex]
So;
Substitute 1 for a and 48 for b in (2) [tex]a + 5b = n[/tex]
[tex]1 + 5 * 48 = n[/tex]
[tex]1 + 240 = n[/tex]
[tex]241 = n[/tex]
[tex]n = 241[/tex]
Hence, the largest value of n is 241
Answer:
Step-by-step explanation:
The two factors of $5x^2+nx+48$ must be in the form $(5x+A)(x+B)$. $A$ and $B$ must be positive integers to form the largest value of $n$. Therefore, $AB=48$ and $5B+A=n$. To form the largest value of $n$, $B$ must equal $48$. Therefore, $A=1$. \[5B+A=5(48)+1=\boxed{241}\]
