2 more questions from the math level 2 SAT subject test! Help please!?

Question

Okay, I've NEVER seen anything like this! If (x - 2) is a factor of (x^3 +kx^2 + 12x - 8), then what is k?

Also, this has to do with inverse functions, and I think I did everything right, but my answer is consistently different than the one provided! If f(x) = (cubed rt)(x^3 + 1), then what is the inverse of f (1.5)?

Any help would be greatly greatly appreciated! Explanations would be great too since I already know the answers... I just don't know how to solve 'em! Thanks so much!

Answer 1

Use the remainder theorem. x - 2 is a factor means f(2) = 0

So, plug in 2 for x and solve for k.

0 = (2)^3 + k(2)^2 + 12(2) - 8
0 = 8 + 4k + 24 - 8
0 = 24 + 4k
-24 = 4k
-6 = k

***************************************
f(x) = cbrt (x^3 + 1)

Find inverse:
y = cbrt (x^3 + 1)
y^3 = x^3 + 1
y^3 - 1 = x^3
cbrt (y^3 - 1) = x
y = cbrt (x^3 - 1)
f inverse = cbrt (x^3 - 1)

Now, plug in 1.5 for x.
f inverse = cbrt [(1.5)^3 - 1]
= cbrt [3.375 - 1]
= cbrt (2.375)
= 1.334200824....

Answer 2

This is an application of remainder theorem.
f(x) = x^3 + kx^2 + 12x - 8
When (x - 2) is a factor then
f(2) = remainder when f(x) is divided by (x - 2) = 0
Now, f(2) = 0 gives
2^3 + k*2^2 + 12*2 - 8 = 0
k = -6



f(x) = (cubed rt)(x^3 + 1)
{f(x)}^3 = (x^3 + 1)
x^3 = {f(x)}^3 - 1
x = (cube rt) [{f(x)}^3 - 1]

Hence, the inverse function is:
(cube rt) [x^3 - 1]
Substituting x = 1.5, you get (cube rt) [1.5^3 - 1] = 1.3342

Answer 3

x^3 +kx^2 +12x - 8
since (x-2) is a factor x = 2 is a solution to the given expression
(2)^3 + k(2)^2 + 12(2) - 8=0
8 +4k +24 - 8 = 0
4k + 24 =0
k = -(24/4) = -6

f(x) = y = (x^3 + 1)^(1/3)
cubing both sides
y^3 = x^3 + 1
x^3 = y^3 -1
x = (y^3 - 1)^(1/3)
so inverse of f(x) = (x^3 - 1)^(1/3)
inverse of f(1.5) = inverse of f(3/2)
= ((3/2)^3 - 1)^(1/3)
= ((27/8) - 1)^(1/3)
= ((27 - 8)/8)^(1/3)
= (19/8)^(1/3)
= 19^(1/3)/2 = (cube root of 19)/2

Answer 4

If (x - 2) is a factor of (x^3 +kx^2 + 12x - 8), then what is k?
x-2=0
x=2

2^3+k 2^2+12. 2 -8=0
8+4k+24-8=0
4k=-24
k=-6

Answer 5

if (x-2) is a factor than f(2)=0=(2^3 +k2^2 + 12*2 - 8)

f(x) = (cubed rt)(x^3 + 1), then what is the inverse of f (1.5)

1.5 = (cubed rt) (x^3 + 1)
3.375=x^3 + 1
2.375=x^3
x = 2.375^(1/3) =1.334200824

to check plug f(1.334200824) = (cubed rt)(1.334200824^3 + 1) = 1.5

Answer 6

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