Given f(x)=4x2−10x−3, according to the Remainder Theorem, f(2) can be found by finding the remainder of which of the following division quotients?
In order to find the remainder of the division, we divide $4x^2 - 10x - 3$ by the polynomial $x-2$.
We can use polynomial long division or synthetic division to do this division. The quotient is $4x-2$ and the remainder is $-7$.
Therefore, according to the Remainder Theorem, $f(2)$ can be found by finding the remainder of the division quotient $-7$.
To find the remainder of a polynomial division, we need to divide the given polynomial f(x) by a linear polynomial of the form (x - c), where c is a constant. In this case, we need to find the remainder of the division of f(x) by (x - 2).
To do the division, we can use long division or synthetic division. Let's use synthetic division:
2 | 4 -10 -3
|______________
To start, we write the coefficients of the polynomial f(x) as the dividend and the constant 2 as the divisor. Then, we place the dividend coefficients under the division symbol and draw a line.
Next, we bring down the first coefficient, which is 4.
2 | 4 -10 -3
|______________
2 | 4 -10 -3
|______________
4
Now, we multiply the divisor (2) by the result we obtained (4) and write the product (-10) under the next coefficient (-10).
2 | 4 -10 -3
|______________
4
-10
Next, we add the values in the second column: -10 + (-10) = -20. We write this sum (-20) under the line.
2 | 4 -10 -3
|______________
4
-10
-20
We repeat the steps above until we reach the last coefficient. Now, we multiply the divisor (2) by the result we obtained (-20) and write the product (-40) under the last coefficient (-3).
2 | 4 -10 -3
|______________
4
-10
-20
-40
Next, we add the values in the second column: -40 + (-3) = -43. We write this sum (-43) under the line.
2 | 4 -10 -3
|______________
4
-10
-20
-40
-43
At this point, we have completed the synthetic division. The remainder is the value at the bottom of the division, which is -43.
Therefore, according to the Remainder Theorem, f(2) can be found by finding the remainder of the division quotient 4x^2 - 10x - 3 divided by (x - 2), which is -43.