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. [tex]3x^{4}-10x^{3} -24x^{2} -6x+5;(x-5)[/tex] as a factor and -1 as a root
Accepted Solution
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Answer:The factors of the polynomial are (x - 5) , (x + 1) , (x + 1) , (3x - 1)The zeroes of it are 5 , -1 , -1 and 1/3Step-by-step explanation:[tex](3x^{4}-10x^{3}-24x^{2}-6x+5)[/tex] ÷ (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 = 5x + 1 = 0 ⇒ x = -13x - 1 = 0 ⇒ 3x = 1 ⇒ x = 1/3The zeroes are 5 , -1 , -1 and 1/3