Question

Use Synthetic Division to Factor the following polynomials completely by given factors or roots and Find all the zeros; When you reach quadratic equation, performance regular factoring or Quadratic Formula.
$3x^{4}-10x^{3} -24x^{2} -6x+5;(x-5)$ as a factor and -1 as a root

Answer 1

Answer:

The factors of the polynomial are (x - 5) , (x + 1) , (x + 1) , (3x - 1)

The zeroes of it are 5 , -1 , -1 and 1/3

Step-by-step explanation:

$(3x^{4}-10x^{3}-24x^{2}-6x+5)$ ÷ (x - 5) =

3x³ + (5x³³ - 24x² - 6x + 5) ÷ (x - 5) =

3x³ + 5x² + (x² - 6x + 5) ÷ (x - 5) =

3x³ + 5x² + x + (-x + 5) ÷ (x - 5) =

3x³ + 5x² + x - 1

∵ x = -1

∴ (x + 1) is a factor

∴ (3x³ + 5x² + x - 1) ÷ (x + 1) = 3x² + (2x² + x - 1) ÷ (x + 1) =

  3x² + 2x + (-x - 1) ÷ (x + 1) = 3x² + 2x - 1

∵ 3x² + 2x - 1 is quadratic so we can factorize it

∴ 3x² + 2x - 1 = (3x - 1)(x + 1)

∴ The factors of the polynomial are (x - 5) , (x + 1) , (x + 1) , (3x - 1)

The zeroes:

x - 5 = 0 ⇒ x = 5

x + 1 = 0 ⇒ x = -1

3x - 1 = 0 ⇒ 3x = 1 ⇒ x = 1/3

The zeroes are 5 , -1 , -1 and 1/3