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).

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

Therefore,

Answer:

The Remainder When p(x)=x^4-2x^3+3x^2-ax+b is Divided by (x – 2) is 10.

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