Question

Obtain all zeros of $f(x)= x^{3} +13x^{2}+32x+20$ if one of its zeros is -2.

Answer 1

Method of finding the remaining zeros of a polynomial when some of its zeros are given:

We firstly write the factor of polynomial using given zeros and multiply them to get g(x). Then divide a given polynomial by g(x).

The quotient so obtained give other zeros of given polynomial and we factorise it to get other zeros.

SOLUTION:

Let f(x) = x³  +13x²  + 32x  +20

Given : -2 is one of the Zeroes of the cubic  polynomial. Therefore ,    

(x + 2) is the factor of given Polynomial f(x).

Now, Divide f(x) = x³  +13x²  + 32x  +20   by g(x) = x + 2.  

[DIVISION IS IN THE ATTACHMENT.]

Hence , all the zeroes of the given Polynomial are: (- 2), (-1) & -10.

HOPE THIS ANSWER WILL HELP YOU …..

Answer 2

let other two zeroes be a and b

sum of three zeroes = ( - coefficient of x^2)/ coefficient of x^3

-2 + a + b= -13
a = -13 +2 - b = -11 - b

Product of three zeroes = - constant / coefficient of x^3

-2 a b = -20

ab = 10

a = 10/b

As a = -11 - b

so compare

10/b = -11 - b

10 = -11b - b^2

b^2 + 11b +10= 0

b^2 + b + 10 b +10= 0

b( b+1) + 10 ( b +1)= 0

b = -10,-1

a = 10/b = -1, -10

so other two zeroes are -1, -10

So all three zeroes are -2, -1, -10

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