Question

If f, left bracket, x, right bracket, equals, 4, x, to the power 5 , plus, 5, x, to the power 4 , plus, 4f(x)=4x
5
+5x
4
+4, then what is the remainder when f, left bracket, x, right bracketf(x) is divided by x, minus, 1x−1?

Answer 1

To find the remainder when f(x)f(x) is divided by x−1x−1, we can use polynomial long division.

Let's divide f(x)f(x) by x−1x−1:
4x^5 + 5x^4 + 4x
---------------------------------
x - 1 | 4x^5 + 5x^4 + 4x

We start by dividing the highest degree terms, which is 4x^5 / x = 4x^4. We write this term above the division line.
4x^4
---------------------------------
x - 1 | 4x^5 + 5x^4 + 4x

Next, we multiply the divisor (x−1x−1) by the quotient we just found (4x^4). 4x^4 * (x−1x−1) = 4x^5 − 4x^4. We write this expression below the polynomial we are dividing.
4x^4
---------------------------------
x - 1 | 4x^5 + 5x^4 + 4x
- (4x^5 - 4x^4)

Now, we subtract the above expression from the polynomial we are dividing:
4x^4
---------------------------------
x - 1 | 4x^5 + 5x^4 + 4x
- (4x^5 - 4x^4)
---------------------------------
9x^4 + 4x

We repeat the above steps by dividing the highest degree term of the new polynomial (9x^4 / x = 9x^3):
4x^4 + 9x^3
---------------------------------
x - 1 | 4x^5 + 5x^4 + 4x
- (4x^5 - 4x^4)
------------------
9x^4 + 4x
9x^4 - 9x^3

We continue this process until the degree of the new polynomial is less than the degree of the divisor x−1x−1.
The final quotient we obtain is 4x^4 + 9x^3. The remainder is 9x^4 + 4x.

Therefore, the remainder when f(x)f(x) is divided by x−1x−1 is 9x^4 + 4x.