Certainly! To find the value of [tex]\( n \)[/tex] such that the polynomial [tex]\( f(x) = 4x^3 + nx^2 + 7x - 23 \)[/tex] when divided by [tex]\( (2x - 5) \)[/tex] leaves a remainder of 7, let's proceed step-by-step:
### Step 1: Understand the problem
The problem states that when [tex]\( f(x) \)[/tex] is divided by [tex]\( (2x - 5) \)[/tex], the remainder is 7. According to the Remainder Theorem, if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( (x - a) \)[/tex], the remainder is [tex]\( f(a) \)[/tex].
### Step 2: Apply the Remainder Theorem
Here, our divisor is [tex]\( 2x - 5 \)[/tex]. We can rewrite this as:
[tex]\[
2x - 5 = 0 \implies x = \frac{5}{2}
\][/tex]
According to the Remainder Theorem:
[tex]\[
f\left(\frac{5}{2}\right) = 7
\][/tex]
We will substitute [tex]\( x = \frac{5}{2} \)[/tex] into the polynomial and set it equal to 7.
### Step 3: Substitute [tex]\( x = \frac{5}{2} \)[/tex] into [tex]\( f(x) \)[/tex]
Substitute [tex]\( x = \frac{5}{2} \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{5}{2}\right) = 4\left(\frac{5}{2}\right)^3 + n\left(\frac{5}{2}\right)^2 + 7\left(\frac{5}{2}\right) - 23
\][/tex]
### Step 4: Simplify the polynomial
Calculate each term:
[tex]\[
\left(\frac{5}{2}\right)^2 = \frac{25}{4}
\][/tex]
[tex]\[
\left(\frac{5}{2}\right)^3 = \frac{125}{8}
\][/tex]
Now substitute back into the polynomial:
[tex]\[
4 \cdot \frac{125}{8} + n \cdot \frac{25}{4} + 7 \cdot \frac{5}{2} - 23 = 7
\][/tex]
Simplify each term:
[tex]\[
4 \cdot \frac{125}{8} = \frac{500}{8} = 62.5
\][/tex]
[tex]\[
n \cdot \frac{25}{4} = \frac{25n}{4}
\][/tex]
[tex]\[
7 \cdot \frac{5}{2} = \frac{35}{2} = 17.5
\][/tex]
### Step 5: Set up the equation
Now we have:
[tex]\[
62.5 + \frac{25n}{4} + 17.5 - 23 = 7
\][/tex]
Combine the constants:
[tex]\[
62.5 + 17.5 - 23 = 57
\][/tex]
So:
[tex]\[
57 + \frac{25n}{4} = 7
\][/tex]
### Step 6: Solve for [tex]\( n \)[/tex]
Isolate [tex]\( n \)[/tex]:
[tex]\[
\frac{25n}{4} = 7 - 57
\][/tex]
[tex]\[
\frac{25n}{4} = -50
\][/tex]
Multiply both sides by 4 to clear the fraction:
[tex]\[
25n = -200
\][/tex]
Now divide by 25:
[tex]\[
n = \frac{-200}{25} = -8
\][/tex]
### Step 7: Verify the answer with the given choices
Given choices are:
A. -10.2
B. -9.6
C. -8.0
D. -7
The computed value [tex]\( n = -8 \)[/tex] matches option C.
### Conclusion
Thus, the correct value of [tex]\( n \)[/tex] is:
[tex]\[
\boxed{-8.0}
\][/tex]
