Question

if 2 zeros of a polynomial p(x)=x4-5x3-x2+35x-42 are 2 and 3 find other 2 zeros of polynomial​

Answer 1

if 2 and 3 are zeros of p(x)

then (x-2),(x-3) are factors of p(x)

now (x-2)(x-3)=0

x^2-5x+6=0

now divide the p(x) with x^2-5x+6

and u will get quotient as x^2-7

x^2-7=0

x^2=7

x=+root7, -root7

these are other 2 zeros of p(x)

Answer 2

Polynomials

Solution:

Given that, 2 and 3 are the zeroes of p(x).

Then (x - 2) and (x - 3) are the factors of p(x).

i.e., (x - 2) (x - 3) is a factor of p(x)

i.e., (x² - 5x + 6) is a factor of p(x)

Now, p(x)

= x⁴ - 5x³ - x² + 35x - 42

= x⁴ - 5x³ + 6x² - 7x² + 35x - 42

= x² (x² - 5x + 6) - 7 (x² - 5x + 6)

= (x² - 5x + 6) (x² - 7)

∴ the other factor of p(x) is (x² - 7).

Now, x² - 7

= x² - (√7)²

= (x + √7) (x - √7)

∴ the other two zeroes are + √7, - √7 .

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If two zeroes of a polynomial p(x) = x⁴ – 5x³ – x² + 35x – 42 are 2 and 3, then find the other two zeroes of the polynomial.

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