DIRECTION III: WORKOUT

36. Let [tex]f(x) = x^3 - 2x^2 + kx + 6[/tex], where [tex]k[/tex] is a constant. If [tex]x - 3[/tex] is a factor of [tex]f(x)[/tex], then factorize [tex]f(x)[/tex] into three linear factors and find the remainder when [tex]f(x)[/tex] is divided by [tex]x + 3[/tex].

37. A function [tex]f(x) = x^3 + x^2 - x + k[/tex], where [tex]k[/tex] is a constant. Given that [tex]x - k[/tex] is a factor of [tex]f(x)[/tex], determine the possible values of [tex]k[/tex].

38. Let [tex]f(x) = ax^3 - x^2 - 5x + b[/tex], where [tex]a[/tex] and [tex]b[/tex] are constants. When [tex]f(x)[/tex] is divided by [tex]x - 2[/tex], the remainder is 36. When [tex]f(x)[/tex] is divided by [tex]x + 2[/tex], the remainder is 40. Find the values of [tex]a[/tex] and [tex]b[/tex].

39. When [tex]f(x) = x^5 - 6x + 2[/tex] is divided by [tex]x - c[/tex], the remainder is 7. Find the possible value of [tex]c[/tex].

40. Let [tex]p(n) = n^3 + 2n^2 - 23n + k[/tex]. If you divide [tex]p(n)[/tex] by [tex]n + 6[/tex], the remainder is -1. What is the remainder if you divide [tex]p(n)[/tex] by [tex]n - 5[/tex]?

41. Find a polynomial [tex]f(x)[/tex] that has roots 0, 1, and -1 with respective multiplicities 3, 2, and 1 such that [tex]f(2) = 24[/tex].

42. Using the remainder theorem, find the remainder when [tex]f(x)[/tex] is divided by [tex]g(x)[/tex].
[tex]f(x) = 4x^3 - 12x^2 + 14x - 3[/tex]; [tex]g(x) = 2x - 1[/tex].

To find the remainder when [tex]$ f(x) = 4x^3 - 12x^2 + 14x - 3 $[/tex] is divided by [tex]$ g(x) = 2x - 1 $[/tex], we can use the Remainder Theorem.

The Remainder Theorem states that if a polynomial [tex]$ f(x) $[/tex] is divided by a linear divisor of the form [tex]$ x - c $[/tex], then the remainder of this division is equal to [tex]$ f(c) $[/tex].

In this case, the divisor [tex]$ g(x) = 2x - 1 $[/tex] can be rewritten in the form of [tex]$ x - c $[/tex] by setting it equal to zero:

[tex]\[ 2x - 1 = 0 \][/tex]

Solving for [tex]$ x $[/tex], we get:

[tex]\[ 2x = 1 \][/tex]

[tex]\[ x = \frac{1}{2} \][/tex]

So, to find the remainder, we evaluate [tex]$ f(x) $[/tex] at [tex]$ x = \frac{1}{2} $[/tex]:

[tex]\[ f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^3 - 12\left(\frac{1}{2}\right)^2 + 14\left(\frac{1}{2}\right) - 3 \][/tex]

Calculating each term:

1. [tex]$ 4\left(\frac{1}{2}\right)^3 = 4 \times \frac{1}{8} = \frac{1}{2} $[/tex]

2. [tex]$- 12\left(\frac{1}{2}\right)^2 = -12 \times \frac{1}{4} = -3$[/tex]

3. [tex]$ 14\left(\frac{1}{2}\right) = 14 \times \frac{1}{2} = 7 $[/tex]

4. [tex]$- 3$[/tex]

Adding these up:

[tex]\[\frac{1}{2} - 3 + 7 - 3 = \frac{1}{2} + 4 - 3 = \frac{1}{2} + 1 = \frac{3}{2} \][/tex]

Thus, the remainder when [tex]$ f(x) $[/tex] is divided by [tex]$ g(x) = 2x - 1 $[/tex] is [tex]$\frac{3}{2}$[/tex].