Question

find all zeros of pollynomial x4 + x3 -34x2-4x+120 if two zeros are 2 and -2​

Answer 1

$\huge\sf\pink{Answer}$

☞ The other two zeros are 5 & -6

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$\huge\sf\blue{Given}$

✭ Two of the zeros of the polynomial x⁴+x³-34x²-4x+120 are 2 & -2

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$\huge\sf\gray{To \:Find}$

◈ All the other zeros of the Polynomial?

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$\huge\sf\purple{Steps}$

☯ $\underline{\textsf{As Per the Question}}$

Two zeros are 2 & -2

➠ $\sf x = 2$

➠ $\sf x-2 = 0$

Similarly

➠ $\sf x = -2$

➠ $\sf x+2 = 0$

Multiplying the both,

➳ $\sf (x-2)(x+2)$

➳ $\sf x^2+2x-2x-4$

➳ $\sf \green{x^2-4}$

So we shall divide x⁴+x³-34x²-4x+120 with x²-4

For division steps take a look at the attachment!!

◕ Quotient = x²+x-30

◕ Remainder = 0

Finding its zeros,

➝ $\sf x^2+x-30$

➝ $\sf x^2+6x-5x-30 = 0$

➝ $\sf x(x+6) -5(x+6) = 0$

➝ $\sf (x-5)(x+6)$

Equating with zero

»» $\sf x-5 = 0$

»» $\sf x = 5$

Similarly,

»» $\sf x+6 = 0$

»» $\sf x = -6$

«« $\sf \orange{Zeros are 5,-6,2 \ \& -2}$ »»

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Answer 2

let the two zeroes of the polynomial be 2 and -2

the factor from this polynomial will be

x=2 and x= -2 = (x-2) . (x+2) = x²+2x-2x-4

= x²- 4 is a factor of polynomial

Now, we divide polynomial by the x²- 4