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?
To find the remainder when f(x)f(x) is divided by x−1x−1, we can use the Remainder Theorem. According to the Remainder Theorem, if you divide a polynomial f(x)f(x) by (x−a)(x−a), the remainder is equal to f(a)f(a), where a is the value subtracted in the denominator.
In this case, we want to find the remainder when f(x)f(x) is divided by x−1x−1. So, we need to find f(1)f(1) since we are subtracting 1 in the denominator.
To find f(1)f(1), we substitute x=1 into the given expression for f(x)f(x):
f(1) = 4(1)^5 + 5(1)^4 + 4 = 4 + 5 + 4 = 13.
Therefore, the remainder when f(x)f(x) is divided by x−1x−1 is 13.
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.
To find the remainder when f(x)f(x) is divided by x−1x−1, we can use the remainder theorem.
The remainder theorem states that if a polynomial f(x)f(x) is divided by x−ax−a, the remainder is equal to f(a)f(a).
In this case, the divisor is x−1x−1. So, we need to find f(1)f(1) to determine the remainder.
To find f(1)f(1), let's substitute x=1 into the given function f(x)f(x):
f(1) = 4(1)^5 + 5(1)^4 + 4f(1)
= 4 + 5 + 4f(1)
= 9 + 4f(1)
Now, we need to find f(1)f(1) using another method. Notice that f(x)f(x) is a quadratic equation in f(x)f(x). So, we can solve the equation f(x)f(x)=4xf(1) + 5x^4 + 4f(x) = 0 to find the value of f(1)f(1).
Let's substitute f(x)=0 into the equation:
0 = 4x + 5x^4 + 4(0)
= 4x + 5x^4
Now, plug in x=1:
0 = 4(1) + 5(1)^4
= 4 + 5
= 9
Therefore, f(1)=9.
So, the remainder when f(x)f(x) is divided by x−1x−1 is equal to f(1)f(1) which is 9.