Question

If two zeroes of the polynomial x4+x3-15x2-29x-6 are 2+square root 5 find other zeroes

Answer 1

two other zeroes are -2 and -3

Two zeroes of the polynomial x⁴ + x³ - 15x² - 29x - 6 are (2 ± √5).

⇒(x - 2 - √5) and (x - 2 + √5) are factors of polynomial,x⁴ + x³ - 15x² - 29x - 6

⇒(x - 2)² - (√5)² is a factor of polynomial, x⁴ + x³ - 15x² - 29x - 6

⇒x² - 4x+ 4 - 5 = x² - 4x - 1 is a factor of polynomial, x⁴ + x³ - 15x² - 29x - 6

now dividing x⁴ + x³ - 15x² - 29x - 6 by x² - 4x- 1.

x² - 4x - 1)x⁴ + x³ - 15x² - 29x - 6(x² + 5x + 6

x⁴ - 4x³ - x²

...............................................

+ 5x³ - 14x² - 29x

+ 5x³ - 20x² - 5x

.........................................................

+ 6x² - 24x - 6

+ 6x² - 24x - 6

.........................................................

0

so another factor is x² + 5x + 6

now solve it,

x² + 5x + 6 = 0

⇒x² + 2x + 3x + 6 = 0

⇒x(x + 2) + 3(x + 2) = 0

⇒(x + 2)(x + 3) = 0

⇒x = -2,-3

hence, two other zeroes are -2 and -3

Answer 2

Step-by-step explanation:

Hope you can understand it