When x 2 is divided into f x, the remainder is 3. Determine the remainder when x 2 is
x 2, the remainder is – 7. What are the values of m and g?
f x is divided by Determine the remainder when x3 3x2 x 2 is divided by x 3x 5.
divided into each of the following:
fx1 b) fxx2 c) fx4x7 d) 2fx7 e) fx2
To determine the remainder when x+2 is divided by f(x), we can write the division as:
x+2 = q(x) * f(x) + 3
where q(x) is the quotient.
To determine the remainder when x+2 is divided by x+2, we can write the division as:
x+2 = q(x) * (x+2) - 7
where q(x) is the quotient.
From the given information, we can set up the following equations:
x+2 = q(x) * f(x) + 3
x+2 = q(x) * (x+2) - 7
Solving these equations simultaneously will give us the values of m and g:
q(x) * f(x) + 3 = q(x) * (x+2) - 7
q(x) * f(x) - q(x) * (x+2) = -10
q(x) (f(x) - (x+2)) = -10
q(x) = -10 / (f(x) - (x+2))
Now, we need more information about f(x) to determine the values of m and g.
To determine the remainder when (x^3 + 3x^2 - x - 2) is divided by (x+3)(x+5), we can use polynomial long division or synthetic division. Since polynomial long division can be lengthy to type, I will use synthetic division:
We set up the division:
-3 -5
---------------
3 | 1 3 -1 -2
-3 0 3
---------------
1 0 2 1
So, the remainder when (x^3 + 3x^2 - x - 2) is divided by (x+3)(x+5) is 2x + 1.
Finally, we can evaluate the expressions:
a) f(x)+1: The remainder is 3 + 1 = 4.
b) f(x)+x+2: The remainder is (2x + 1) + x + 2 = 3x + 3.
c) f(x)+(4x+7): The remainder is (2x + 1) + (4x + 7) = 6x + 8.
d) 2f(x)-7: The remainder is 2(2x + 1) - 7 = 4x - 5.
e) [f(x)]^2: The remainder is (2x + 1)^2 = 4x^2 + 4x + 1.
To determine the remainder when x + 2 is divided into f(x), we can use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by x - c, the remainder is equal to f(c).
Given that the remainder is 3 when x + 2 is divided into f(x), it means that f(-2) = 3.
Similarly, if the remainder is -7, it means that f(-2) = -7.
Since f(-2) cannot be both 3 and -7 at the same time, there seems to be a contradiction. Please double-check the given information and provide the correct values.
Moving on to the second part of the question, we need to find the remainder when (x^3 + 3x^2 - x - 2) is divided by (x + 3)(x + 5).
To find the remainder, we can use long division or synthetic division method. Let's use synthetic division.
We divide (x^3 + 3x^2 - x - 2) by (x + 3):
-3 | 1 3 -1 -2
| -3 0 3
_______________________
1 0 2 1
The remainder is 1.
Now, we divide the obtained quotient (x^2 + 2) by (x + 5):
-5 | 1 0 2
| -5 25 -115
___________________
1 -5 27
The remainder is 27.
Therefore, when (x^3 + 3x^2 - x - 2) is divided by (x + 3)(x + 5), the remainder is 27.
For the next parts of the question:
a) f(x) + 1: The remainder will remain the same because adding 1 to f(x) will not affect the divisibility by x + 2.
b) f(x) + x + 2: The remainder will remain the same because adding x + 2 to f(x) will not affect the divisibility by x + 2.
c) f(x) + (4x + 7): The remainder will remain the same because adding (4x + 7) to f(x) will not affect the divisibility by x + 2.
d) 2f(x) - 7: The remainder will remain the same because multiplying f(x) by 2 and subtracting 7 from it will not affect the divisibility by x + 2.
e) [f(x)]^2: There is no specific rule to determine the remainder when f(x) is squared. We would need additional information about f(x) to proceed.
Please provide further instructions or clarify any doubts if needed.
To determine the remainder when dividing polynomials, we can use polynomial long division or synthetic division. I will explain each method briefly, and then we can apply them to solve the given problems.
1. Polynomial Long Division: This method involves dividing the polynomials term by term. It is similar to long division for numbers. It requires arranging the terms in decreasing order of powers, dividing the highest degree terms, multiplying the result by the divisor, subtracting, and repeating the process with the next term until all terms are exhausted. The remainder is the final result.
2. Synthetic Division: This method is a shorthand version of polynomial long division, specifically for dividing by a linear divisor of the form (x+c). It simplifies the process by only considering the coefficients and requires fewer steps. Synthetic division can only be used when dividing by linear divisors.
Now, let's solve the given problems using the above methods:
Problem 1: When x+2 is divided into f(x), the remainder is 3. We need more information to determine the values of m and g since the question is incomplete.
Problem 2: When x+2 is divided into f(x), the remainder is -7. We need more information to determine the values of m and g since the question is incomplete.
Problem 3: To determine the remainder when (x^3 + 3x^2 - x - 2) is divided by (x + 3)(x + 5), we can use polynomial long division. Arrange the terms in descending order of powers:
____________
(x + 3)(x + 5) | x^3 + 3x^2 - x - 2
Perform the division step by step, starting with the highest degree term:
-x^2 + 6x - 17
__________________________
(x + 3)(x + 5) | x^3 + 3x^2 - x - 2
-(x^3 + 3x^2)
________________
-4x^2 - x
-(-4x^2 - 12x)
______________
11x - 2
-(11x + 33)
______________
-35
The remainder is -35.
Problem 4: a) To determine the remainder when f(x) is divided by f(x) + 1, we need more information about the function f(x). Without knowing the specific form of f(x), it is not possible to find the remainder.
Problem 4: b) To determine the remainder when f(x) is divided by f(x) + x + 2, we need more information about the function f(x). Without knowing the specific form of f(x), it is not possible to find the remainder.
Problem 4: c) To determine the remainder when f(x) is divided by f(x) + (4x + 7), we need more information about the function f(x). Without knowing the specific form of f(x), it is not possible to find the remainder.
Problem 4: d) To determine the remainder when 2f(x) is divided by 2f(x) - 7, we need more information about the function f(x). Without knowing the specific form of f(x), it is not possible to find the remainder.
Problem 4: e) To determine the remainder when [f(x)]^2 is divided by [f(x)]^2, we need more information about the function f(x). Without knowing the specific form of f(x), it is not possible to find the remainder.