If k is a constant, what is the value of k such that the polynomial k^2x^3+14kx+49 divisible by x-1

To find the value of k such that the polynomial is divisible by x-1 , we can use the Remainder Theorem.

According to the Remainder Theorem, if a polynomial f(x) is divided by x - c , the remainder will be equal to f(c) . In other words, if f(c) = 0 , then x - c is a factor of the polynomial.

In this case, we want the polynomial k^2x^3 + 14kx + 49 to be divisible by x - 1 . This means that when we substitute x = 1 into the polynomial, the result should be equal to zero.

Substituting x = 1 into the polynomial, we have:
k^2(1)^3 + 14k(1) + 49 = 0

Simplifying the equation, we get:
k^2 + 14k + 49 = 0

Now, we can solve this quadratic equation for k. We can either factor the equation or use the quadratic formula.

Using the quadratic formula, the solutions for k are:
k=\frac{-b±\sqrt{b^2-4ac}}{2a}

For this equation, a = 1, b = 14, and c = 49. Substituting these values into the quadratic formula, we get:
k=\frac{-14±\sqrt{14^2-4(1)(49)}}{2(1)}

Simplifying the equation further, we have:
k=\frac{-14±\sqrt{196-196}}{2}

Since the discriminant is equal to zero (196 - 196 = 0), we have only one solution:
k = -7

Therefore, the value of k such that the polynomial k^2x^3 + 14kx + 49 is divisible by x - 1 is k = -7 .

Answer: k = -7