Question

if the zeros of polynomial x³-6x²-8x+24 is in the form of a-b , a and a+b then find all zeros can anybody answer this question if you are brilliant then answer this​

Answer 1

The zero of a polynomial is a value of the variable at which the polynomial equals 0 .

For the said polynomial, we have these equations:

1..

a3−3a2+a+1=0

2...

(a−b)3−3(a−b)2+a−b+1=0

or,a3−3a2b+3ab2−b3−3a2+6ab−3b2+a−b+1=0

3...

(a+b)3−3(a+b)2+a+b+1=0

or,a3+3a2b+3ab2+b3−3a2−6ab−3b2+a+b+1=0

Using equations 1 & 2, we get,

−3a2b+3ab2−b3−3b2+6ab−b=0

or, 3a2b−3ab2+b3+3b2−6ab+b=0

Using equations 1 & 3, we get,

3a2b+3ab2+b3−3b2−6ab+b=0

Subtracting last two equations, we get,

6ab2−6b2=0

or, 6b2(a−1)=0

or, b=0, or a=1

Now, putting a=1 in either equation 1 or equation 2, we get,

3b+3b2+b3−3b2−6b+b=0

or, b3−2b=0

or, b(b2−2)=0

or, b=0 , or b √2 or b= -√2

So, we have got a=1 and b = 0, √2 nd -√2

Therefore, the zeroes of the said polynomial are

1,1 + √2 and 1−√2

.

Answer 2

Question :

If the zeroes of the polynomial x³-6x²-8x+24 is in the form of α-β , α , α+β then find all zeroes

Solution :

We are given a cubic polynomial x³-6x²-8x+24 and roots/zeroes of the poly. is α-β , α , α+β

Compare given cubic poly. with ax³+bx²+cx+d ,

• a = 1  ,  b = -6  ,  c = -8  ,  d = 24

Sum of zeroes of the poly. is ,

$:\implies \sf{(\alpha -\beta)+\alpha+(\alpha+\beta)=-\dfrac{b}{a}}$

$:\implies \sf 3\alpha=-\dfrac{(-6)}{1}$

$\boldsymbol{:\implies} \textsf{\textbf{ \boldsymbol{\alpha=}\ \text{2}}}\ \; \pink{\bigstar}$

Product of zeroes of the poly. is ,

$:\implies \sf (\alpha-\beta)(\alpha)(\alpha+\beta)=-\dfrac{d}{a}$

• ( a + b ) ( a - b ) = a² - b²

• α = 2

$:\implies \sf 2(\alpha^2-\beta^2)=-\dfrac{24}{1}$

$:\implies \sf ((2)^2-\beta^2)=-12$

$:\implies \sf \beta^2=16$

$\textsf{\textbf{ \boldsymbol{:\implies} \boldsymbol{\beta =\pm} \text{4}}}\ \; \orange{\bigstar}$

So , Zeroes of the polynomial are as follows ,

(α-β) = (2-4) = -2 or (α-β) = (2-(-4)) = 6

• α = 2

(α+β) = (2+4) = 6 or (α+β) = (2+(-4)) = -2

-2 , 2 , 6 are zeroes of the polynomial

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