If remainder on division x^3-kx^2+13x-21 by 2x-1 is -21 ,find quotient and k.Hence find zeroes of cubic polynomial x^3 -kx^2+13x-21.
To find the quotient and the value of k, we will make use of polynomial division. Here's how you can proceed:
Step 1: Write the polynomial division equation
Divide x^3 - kx^2 + 13x - 21 by 2x - 1, and the remainder should be -21. Write the equation as follows:
(x^3 - kx^2 + 13x - 21) / (2x - 1) = quotient + (-21 / (2x - 1))
Step 2: Perform polynomial division
Perform the polynomial division to find the quotient and remainder:
We start by dividing the highest degree term. In this case, divide x^3 by 2x, which gives us (1/2)x^2.
Multiply the divisor (2x - 1) by (1/2)x^2, giving us (1/2)x^3 - (1/2)x^2.
Subtract this from the original polynomial to eliminate the highest degree term:
x^3 - (1/2)x^3 = (1/2)x^2.
Bring down the next term, -kx^2, and repeat the process:
(1/2)x^2 - -kx^2 = (k - 1/2)x^2.
Divide ((k - 1/2)x^2 + 13x) by (2x - 1) once again.
Repeat these steps until you reach the end of the polynomial.
At the end of the division, your quotient should be ((k - 1/2)x^2 + 13x - 21) / (2x - 1), and the remainder should be -21.
Step 3: Equate the remainder to -21
From the given information, it is mentioned that the remainder is -21. Hence, equate the remainder to -21:
-21 = -21
Step 4: Solve for k
To find the value of k, equate the coefficients of the highest degree terms in the remainder equation and the given cubic polynomial:
-1/2 = -k
Solving for k, we find:
k = 1/2
Step 5: Find the zeroes of the cubic polynomial
To find the zeroes of the cubic polynomial x^3 - kx^2 + 13x - 21, substitute the value of k (k = 1/2) into the polynomial:
x^3 - (1/2)x^2 + 13x - 21
Using a numerical method like the Newton-Raphson method or a graphing calculator, you can find the zeroes of the polynomial to get the solutions for x.
Just do the long division, and you see the remainder is
(55-4k)/8
So, set that = -21 and solve for k.