Answer :
Answer:
Step-by-step explanation:
Certainly! To find the value of \(k\) in the equation \(x^3 + 6x^2 + 11x + 6 = k(x + 1)\), you can perform polynomial division or use other methods to equate the coefficients of corresponding powers of \(x\) on both sides of the equation.
Given the equation \(x^3 + 6x^2 + 11x + 6 = k(x + 1)\), let's start by expanding \(k(x + 1)\) on the right-hand side:
\[k(x + 1) = kx + k\]
Now, set the coefficients of \(x^3\), \(x^2\), \(x\), and the constant term equal on both sides of the equation:
For \(x^3\):
The coefficient on the left side is \(1\), and there's no \(x^3\) term on the right side, so \(0 = 0\) (as there's no \(x^3\) term on the right).
For \(x^2\):
The coefficient on the left side is \(6\), and on the right side, it should be \(0\) (as there's no \(x^2\) term on the right). So, \(6 = 0\), which is not possible.
Hence, there's no value of \(k\) that satisfies this equation because the coefficients of the terms with \(x^2\) do not match. The equation \(x^3 + 6x^2 + 11x + 6 = k(x + 1)\) is not consistent for any value of \(k\).
Step-by-step explanation:
x^3+6x^2+11x+6=k(x+1)
k=x^3+6x^2+11x+6÷x+1
k=x^2+5x+6