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:
[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 = 5
x + 1 = 0 ⇒ x = -1
3x - 1 = 0 ⇒ 3x = 1 ⇒ x = 1/3
The zeroes are 5 , -1 , -1 and 1/3
