i'm really not sure how to do this question.
(3x+2k)^3+(4x-7)^2 has a remainder of 33 when divided by x-3. find k.
i tried to expand the expression, but i don't get a polynomial, so how do i apply the remainder theorem? i would really appreciate it if anyone could tell me how to solve this question! thanks!
You do not have to expand it,
the remainder theorem would say here that
f(3) = 33, so
(9+2k)^3 + (12-7)^2 = 33
(9+2k)^3 = 8 , take the cube roo
9+2k = 2
2k = -7
k = -7/2
8 DIVIDED BY 200
To solve this question, we'll start by applying the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by x - c, the remainder is equal to f(c). In this case, the polynomial is (3x + 2k)^3 + (4x - 7)^2, and we want to find the value of k such that the remainder is 33 when divided by x - 3.
Step 1: Apply the Remainder Theorem
To find the remainder, we substitute x = 3 into the polynomial and set it equal to 33:
((3(3) + 2k)^3 + (4(3) - 7)^2) = 33.
Step 2: Simplify the equation
Simplifying the equation, we get:
((9 + 2k)^3 + (12 - 7)^2) = 33.
(9 + 2k)^3 + 5^2 = 33.
(9 + 2k)^3 + 25 = 33.
Step 3: Solve for k
Subtracting 25 from both sides of the equation, we have:
(9 + 2k)^3 = 33 - 25.
(9 + 2k)^3 = 8.
Taking the cube root of both sides:
∛((9 + 2k)^3) = ∛8.
Simplifying, we get:
9 + 2k = 2.
Subtracting 9 from both sides:
2k = 2 - 9.
2k = -7.
Dividing both sides by 2, we find:
k = -7/2.
Therefore, the value of k that satisfies the condition is k = -7/2.
To apply the remainder theorem in this case, you first need to expand the given expression. Let's start by expanding (3x+2k)^3:
(3x+2k)^3 = (3x+2k)(3x+2k)(3x+2k)
= (9x^2+12xk+4k^2)(3x+2k)
= 27x^3 + 54x^2k + 36xk^2 + 8k^3
Now, let's expand (4x-7)^2:
(4x-7)^2 = (4x-7)(4x-7)
= 16x^2 - 56x + 49
Now, adding these two expanded expressions together:
(27x^3 + 54x^2k + 36xk^2 + 8k^3) + (16x^2 - 56x + 49)
= 27x^3 + 54x^2k + 16x^2 + 36xk^2 - 56x + 8k^3 + 49
To find the remainder when this expression is divided by x-3, we can substitute x with 3 in the expression and solve for the value of k.
Now, substitute x with 3:
27(3)^3 + 54(3)^2k + 16(3)^2 + 36(3)k^2 - 56(3) + 8k^3 + 49
= 27(27) + 54(9)k + 16(9) + 36(3)k^2 - 168 + 8k^3 + 49
= 729 + 486k + 144 + 108k^2 - 168 + 8k^3 + 49
= 8k^3 + 108k^2 + 486k + 755
Given that this expression has a remainder of 33 when divided by x-3, we can set it equal to 33:
8k^3 + 108k^2 + 486k + 755 = 33
Simplifying the equation by subtracting 33 from both sides:
8k^3 + 108k^2 + 486k + 722 = 0
Unfortunately, finding the value of k in this equation requires solving a cubic equation, which can be complex. You may need to use numerical methods or a graphing calculator to approximate the solution.
Therefore, to find the value of k in this question, you need to solve the equation 8k^3 + 108k^2 + 486k + 722 = 0 using numerical methods or a graphing calculator.