Answer:
k ≈ -53.802
approximately -53.802
Explanation:
To find the value of k for the polynomial expression (7x^3 + kx^2 - 83x + 10)(x - 9) to have a remainder of -8 when divided by (x - 9), you can use the Remainder Theorem. The Remainder Theorem states that the remainder when a polynomial is divided by (x - a) is equal to the value of the polynomial when you substitute "a" into it.
In this case, "a" is 9, so you need to substitute x = 9 into the polynomial:
(7x^3 + kx^2 - 83x + 10)(x - 9) when x = 9
(7(9^3) + k(9^2) - 83(9) + 10)(9 - 9)
Now, calculate the expression:
(7(729) + k(81) - 83(9) + 10)(0)
(5103 + 81k - 747 + 10)(0)
(4356 + 81k)(0)
Since the remainder is given as -8, we can set this equal to -8:
4356 + 81k = -8
Now, solve for k:
81k = -8 - 4356
81k = -4364
k = -4364 / 81
k ≈ -53.802
So, the value of k that makes the polynomial expression have a remainder of -8 when divided by (x - 9) is approximately -53.802.