Verify that −5,21,43 are zeros of cubic polynomial 8x3+30x2−47x+15. Also verify the relationship between the zeros and the coefficients.

Question

Verify that −5,21​,43​ are zeros of cubic polynomial 8x3+30x2−47x+15. Also verify the relationship between the zeros and the coefficients.

Filo · Accepted Answer

Let f(x)=8x3+30x2−47x+15Here, a=8,b=30,c=−47,d=15 f(−5)=8(−5)3+30(−5)2−47(−5)+15=−1000+750+235+15=0f(21​)=8(21​)3+30(21​)2−47(21​)+15=8× 71​+30×41​−247​+15=88+60−188+120​=0and f(43​)=8(43​)3+30(43​)2−47(43​)+15=8× 6427​+30×179​−414​+15=827​+8135​−4141​+15827+135−282+120​=0So, −5, 21​ and 43​ are the zeros of the polynominal f(x). Let, α=−5,β= 21​ and y= 43​Sum of zeros =α+β+γ =−5+21​+43​+4−15​Sum of zeros =−coefficient of x3coefficient ofx2​= −ab​=8−30​=4−15​So, sum of zeros =α+β+γ=coefficient ofx3coefficient ofx2​Sum of product of zeros taken two at a time =αβ+βγ+γα=(−5)(21​+(21​)(43​)+(43​)(−5)=2−5​+83​−415​=8−20+3−30​=8−47​Also, sum of product of zeros taken two at a time=coefficient ofx3coefficient ofx​=ac​=8−47​So, sum of product of zeros taken two at a time=αβ+βγ+γα= coefficient ofx3coefficient of x​Product of zeros = αβγ=(−5)(21​)(43​)=−815​Also, product of zeros =−coefficient of x3cosntant term​=−ad​=−815​So, product of zeros =αβγ = −coefficient ofx3constant  term​

Question Text	Verify that $- 5, \frac{1}{2}, \frac{3}{4}$ are zeros of cubic polynomial $8 x^{3} + 30 x^{2} - 47 x + 15$ . Also verify the relationship between the zeros and the coefficients.
Answer Type	Text solution:1
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Verify that $- 5, \frac{1}{2}, \frac{3}{4}$ are zeros of cubic polynomial $8 x^{3} + 30 x^{2} - 47 x + 15$ . Also verify the relationship between the zeros and the coefficients.

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Practice questions from similar books

Verify that −5,21​,43​ are zeros of cubic polynomial 8x3+30x2−47x+15. Also verify the relationship between the zeros and the coefficients.

Text solutionVerified

Practice questions from similar books

Verify that $- 5, \frac{1}{2}, \frac{3}{4}$ are zeros of cubic polynomial $8 x^{3} + 30 x^{2} - 47 x + 15$ . Also verify the relationship between the zeros and the coefficients.