  # The Polynomial p(x)=x^4-2x^3+3x^2-ax+b when Divided by (x-1) and (x +1) leaves the Remainders 5 and 19 respectively. Find the Values of a and b. Hence, Find the Remainder When p(x) is Divided by (x -2).

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Updated on: October 17th, 2023 Remainder is 10.

Given that p(x) = x^4 – 2x^3 + 3x^2 – ax + b, we know that when p(x) is divided by (x – 1), the remainder is 5, and when divided by (x + 1), the remainder is 19.

In order to solve the given problem, we will use the following steps:

1. Step 1: Divide p(x) by (x – 1) and set the remainder equal to 5. Substitute x = 1 into p(x) and set it equal to 5.
2. Step 2: Simplify the equation obtained from Step 1.
3. Step 3: Divide p(x) by (x + 1) and set the remainder equal to 19. Substitute x = -1 into p(x) and set it equal to 19.
4. Step 4: Simplify the equation obtained from Step 3.
5. Step 5: Solve the system of equations formed by Equation 1 and Equation 2. Add Equation 1 and Equation 2 to eliminate variable b.
6. Step 6: Substitute the value of b into Equation 1 and solve for a.
7. Step 7: Determine the values of a and b.
8. Step 8: Substitute x = 2 into p(x) to find the remainder when divided by (x – 2).

Table of content ## The Polynomial p(x)=x^4-2x^3+3x^2-ax+b when Divided by (x-1) and (x +1) leaves the Remainders 5 and 19 respectively. Find the Values of a and b. Hence, Find the Remainder When p(x) is Divided by (x -2).

Solution:

Dividing p(x) by (x – 1) gives us a remainder of 5. So, substitute x = 1 into p(x) and set it equal to 5:

p(1) = 1^4 – 2(1)^3 + 3(1)^2 – a(1) + b = 5

Simplifying this equation: 1 – 2 + 3 – a + b = 5 -a + b + 2 = 5 -a + b = 3 (Equation 1)

Dividing p(x) by (x + 1) gives us a remainder of 19. So, substitute x = -1 into p(x) and set it equal to 19:

p(-1) = (-1)^4 – 2(-1)^3 + 3(-1)^2 – a(-1) + b = 19

Simplifying this equation: 1 + 2 + 3 + a + b = 19 a + b = 13 (Equation 2)

Now we have a system of two equations (Equation 1 and Equation 2) with two variables (a and b). We can solve this system of equations to find the values of a and b.

Solving Equation 1 and Equation 2 simultaneously, we get:

-a + b = 3

a + b = 13

Adding the two equations, we eliminate the variable b:

2b = 16

b = 8

Substituting the value of b into Equation 1, we can solve for a:

-a + 8 = 3

-a = -5

a = 5

So, the values of a and b are a = 5 and b = 8.

To find the remainder when p(x) is divided by (x – 2), we substitute x = 2 into p(x): p(2) = 2^4 – 2(2)^3 + 3(2)^2 – 5(2) + 8 = 16 – 16 + 12 – 10 + 8 = 10

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