Question

p(x) = 3 (xx+10x+5) - 5 (x-k). In the polynomial p(x) defined, k is a constant. If p(x) is divisible by x, what is the value of k?

Answer 1

To ensure the polynomial p(x) = 3 (x^2+10x+5) - 5 (x-k) is divisible by x, the constant term must be zero. Solving 15 + 5k = 0 gives us k = -3. Hence, the value of k is -3.

Step-by-step explanation:

To find the value of k for which the polynomial p(x) is divisible by x, we need to ensure that there is no constant term when the polynomial is simplified. The given polynomial is:

p(x) = 3 (x2+10x+5) - 5 (x-k)

Expanding this, we get:

p(x) = 3x2 + 30x + 15 - 5x + 5k

To be divisible by x, the constant term in the polynomial must be zero. Combining like terms gives us:

p(x) = 3x2 + (30x - 5x) + (15 + 5k)

p(x) = 3x2 + 25x + (15 + 5k)

Setting the constant term (15 + 5k) equal to zero gives us:

15 + 5k = 0

Now we solve for k:

5k = -15

k = -15 / 5

k = -3

Therefore, the value of k for which the polynomial is divisible by x is -3.