Use the Remainder Theorem to find the value of f(1/2) if f(x)=1/4x^3−1/8x+3 .(1 point)
1. 3
2. 95/32
3. 19/4
4. 25/8
To find the value of f(1/2), we need to plug in x = 1/2 into the polynomial f(x).
f(x) = 1/4x^3 - 1/8x + 3
f(1/2) = 1/4(1/2)^3 - 1/8(1/2) + 3
= 1/4(1/8) - 1/16 + 3
= 1/32 - 1/16 + 3
= 1/32 - 2/32 + 3
= -1/32 + 3
= (3 - 1)/32
= 2/32
= 1/16
Therefore, the value of f(1/2) is 1/16.
So, the correct answer is:
2. 1/16
To find the value of f(1/2) using the Remainder Theorem, you need to divide the polynomial f(x) by x - 1/2 and evaluate the resulting expression at x = 1/2.
First, let's divide the polynomial f(x) = 1/4x^3 - 1/8x + 3 by x - 1/2:
Using long division or synthetic division, the division can be done as follows:
1/4 | 1/4x^3 - 1/8x + 3
- (1/4x^3 - 1/8x^2)
----------------
(1/8x^2 - 1/8x)
- (1/8x^2 - 1/16x)
-----------------
(1/16x + 3)
- (1/16x + 3/32)
-----------------
0
The remainder is zero, indicating that (x - 1/2) is a factor of f(x).
Next, to find f(1/2), substitute x = 1/2 into the expression we obtained after division:
f(1/2) = 1/16(1/2) + 3/32
= 1/32 + 3/32
= (1 + 3)/32
= 4/32
= 1/8
Therefore, the value of f(1/2) is 1/8.
To use the Remainder Theorem to find the value of f(1/2), first, we need to understand what the Remainder Theorem is.
The Remainder Theorem states that if a polynomial f(x) is divided by a linear binomial (x - k), then the remainder is equal to f(k). In other words, if we substitute the value of k into the polynomial, the resulting value is the remainder.
In this case, we want to find the value of f(1/2) for the polynomial f(x) = 1/4x^3 - 1/8x + 3.
To find f(1/2), we substitute x = 1/2 into the polynomial:
f(1/2) = 1/4(1/2)^3 - 1/8(1/2) + 3
Simplifying this expression:
f(1/2) = 1/4(1/8) - 1/8(1/2) + 3
f(1/2) = 1/32 - 1/16 + 3
Now, let's add the fractions to obtain a common denominator:
f(1/2) = 1/32 - 2/32 + 96/32
f(1/2) = (1 - 2 + 96)/32
f(1/2) = 95/32
Therefore, the value of f(1/2) is 95/32. Hence, the correct option is 2.