Given f(x)=4x^2-10x-3, according to the remainder theorem, f(2) can be
calculated by substituting x=2 into the function f(x).
Let's substitute x=2 into the function f(x):
f(2) = 4(2)^2 - 10(2) - 3
= 4(4) - 20 - 3
= 16 - 20 - 3
= -4 - 3
= -7
Therefore, according to the remainder theorem, f(2) equals -7.
calculated by dividing the polynomial f(x) by (x - 2) and evaluating the remainder.
To find the remainder, we need to perform polynomial division:
2x - 1
-----------------
x - 2 | 4x^2 - 10x - 3
- (4x^2 - 8x)
--------------
- 2x - 3
+ ( - 2x + 4)
--------------
1
The remainder is 1, so according to the remainder theorem, f(2) = 1.
calculated by finding the remainder when dividing the polynomial f(x) by (x - 2). To use the remainder theorem, we first need to find the quotient and remainder of the division.
To do this, we can use long division or synthetic division. Let's use synthetic division in this case:
1. Write down the coefficients of f(x) in descending order: 4, -10, -3. If there are any missing terms, fill them with zeros.
2. Next, write down the divisor: x - 2.
3. Since the divisor is in the form (x - a), we change the sign of "a" and write it next to the divisor. In this case, it is -2.
4. Start the synthetic division process by bringing down the first coefficient (4):
4 | 4 -10 -3
______________
Bring down the 4.
4 | 4 -10 -3
_________
4
5. Multiply the divisor (-2) by the number at the bottom (4):
4 | 4 -10 -3
_________
4
-8
6. Add the result (-8) to the next coefficient (-10) and write it below:
4 | 4 -10 -3
_________
4
-8
------
-18
7. Repeat steps 5 and 6 until you reach the last coefficient (-3). In this case, there is no more coefficient to bring down.
4 | 4 -10 -3
_________
4
-8
------
-18
16
--------
-2
8. The number at the bottom, -2, is the remainder.
Now we can use the remainder theorem to find f(2). According to the remainder theorem, if we substitute the divisor (-2) into the polynomial, the result should be equal to the remainder.
f(2) = -2
Therefore, according to the remainder theorem, f(2) can be -2.