When 3x^2 + 10x - 3 is divided by x+k, the remainder is 5. What is k?

Question

It is polynomial functions. PLEASE HELP!!! ASAP!!!!!!!!! THANK YOU.

Answer 1

From Remainder Theorem: When (x - a) is divided into polynomial P(x), remainder is P(a).

 

So, P(-k) is 5.  Solve 3x^2 + 10 x - 3 = 5:

 

3x^2 + 10x - 8 = 0

 

Find two factors of (3)(-8) = -24 that add up to 10:

 

2 and -12 work, so this is:

 

(3x - 2)(x + 4) = 0

 

so for one of these to be zero, x has to equal 2/3 or -4.  This means that P(2/3) = P(-4) = 5.  By inspection, k = 4.

 

 

 

                     

Answer 2

By the Remainder Theorem, when a polynomial function f(x) is divided by x - k, the remainder is f(k).  

 

Let f(x) = 3x² + 10x - 3

 

x + k = x - (-k)

 

By the Remainder Theorem, f(-k) = 5

 

So, 3(-k)² +10(-k) - 3 = 5

 

     3k² - 10k - 8 = 0

 

    (3k + 2)(k - 4) = 0    k = -2/3 or k = 4

 

 

Answer 3

Using the Remainder Theorem:

When a polynomial P(x) is divided by (x + k), the remainder is P(-k).

So I need: P(-k) = 5

Setting up the equation:

P(x) = 3x² + 10x - 3

P(-k) = 3(-k)² + 10(-k) - 3 = 5

Solving:

3k² - 10k - 3 = 5

3k² - 10k - 8 = 0

Using the quadratic formula:

k = [10 ± √(100 + 96)] / 6

k = [10 ± √196] / 6

k = [10 ± 14] / 6

Two solutions:

k = (10 + 14)/6 = 24/6 = 4

or

k = (10 - 14)/6 = -4/6 = -2/3

Therefore, k = 4 or k = -2/3