Question

Given p(x) = x 4 + x 3 - 13x 2 - 25x - 12 What is the remainder when p(x) is divided by x - 4? Describe the relationship between the linear expression and the polynomial?

Answer 1

A: x² - 9 with a remainder of -12x - 48

B: Shown below

Explanation:

A: (x⁴ – 13x² – 25x – 12) ÷ (x² - 4)

x² goes into x⁴, x² times

x⁴ – 13x² – 25x – 12

-(x⁴ - 4x²)

= -9x²

From here, the next two terms are brought down

-9x² - 25x - 12

x² goes into this expression -9 times

-9x² - 25x - 12

(-9x² - 0x + 36)

= -12x - 48

So the answer is x² - 9 with a remainder of -12x - 48

B: To prove the answer as correct, the answer must be multiplied by the expression (x² - 4)

(x² - 9)(x² - 4) - 25x - 48

x⁴ - 4x² - 9x² + 36 - 25x - 48

x⁴ - 13x² - 25x + 36 - 48

x⁴ - 13x² - 25x - 12 = p(x)

Answer 2

I'm assuming the function is p(x) = x^4 + x^3 - 13x^2 - 25x - 12

If so, then we can plug in 4 to get

p(x) = x^4 + x^3 - 13x^2 - 25x - 12
p(4) = (4)^4 + (4)^3 - 13(4)^2 - 25(4) - 12
p(4) = 256 + 64 - 13(16) - 25(4) - 12
p(4) = 256 + 64 - 208 - 100 - 12
p(4) = 320 - 208 - 100 - 12
p(4) = 112 - 100 - 12
p(4) = 12 - 12
p(4) = 0

Since the result is 0, this means that x-4 is a factor of p(x). This is due to the remainder theorem.