Question

How many real roots does 2+7x+14=0" alt="x^4-x^3-9x^2+7x+14=0" align="absmiddle" class="latex-formula"> have?
A) 1
B) 2
C) 3
D) 4

Answer 1

Answer: D) 4

Step-by-step explanation:

x⁴ - x³ - 9x² + 7x + 14 = 0

possible rational roots are: ±{1, 2, 7, 14}

Try x = -1

-1  |   1   -1   -9   7   14

   |   ↓  -1     2   7   -14

       1   -2    -7   14   0 ← remainder is 0 so x = -1 is a root

Let's try x = 2 with the factored polynomial

2  |  1   -2   -7   14

   |  ↓    2   0   -14

       1    0   -7    0  ← remainder is 0 so x = 2 is a root

Factored polynomial x² - 7 can be factored to:

(x - √7) (x + √7)

⇒ x = √7   and x = -√7

The roots are x = {-1, 2, √7, -√7) which are all real numbers.