Given the zeroes of a cubic polynomial fx-2x3-x2-5x-2- 1-2-1-2 Verify the relation between its zeros and coefficients-

Therefore, nbsp;α nbsp;= nbsp;1/2 nbsp;β nbsp;= 1, nbsp;γ nbsp;= 2 now, nbsp; Sum of roots = nbsp;

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Standard X

Mathematics

Relationship between Zeroes and Coefficients of a Polynomial

Given the zer...

Question

Given the zeroes of a cubic polynomial $f (x) = 2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$ Verify the relation between its zeros and coefficients.

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Solution

Therefore, $α = \frac{1}{2} β = 1, γ = - 2$
now,
Sum of roots = $fraction numerator negative left parenthesis c o e f f i c i e n t space o f space x Â² right parenthesis over denominator left parenthesis c o e f f i c i e n t space o f space x Â³ right parenthesis end fraction$
$fraction numerator negative space left parenthesis 1 right parenthesis space over denominator 2 end fraction equals space 1 half space plus space 1 space minus 2 space space L H S space equals R H S$

Products of roots = $fraction numerator negative left parenthesis space c o n s tan t space right parenthesis over denominator c o e f f i c i e n t space o f space x Â³ end fraction$
$fraction numerator negative space left parenthesis 2 right parenthesis space over denominator 2 end fraction equals space 1 half space cross times space 1 space cross times space minus 2 space minus 1 equals negative 1 L H S equals R H S$
Sum of products of two consecutive roots = $fraction numerator left parenthesis space c o e f f i c i e n t space o f space x right parenthesis over denominator c o e f f i c i e n t space o f space x Â³ end fraction$
$fraction numerator negative 5 over denominator 2 end fraction equals space 1 half space cross times space 1 space plus space 1 space cross times space left parenthesis negative 2 right parenthesis space plus space left parenthesis negative 2 right parenthesis space cross times space 1 half space space fraction numerator negative 5 over denominator 2 end fraction equals space fraction numerator negative 5 over denominator 2 end fraction$
LHS=RHS

Hence Verified

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Similar questions

Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in in each case:

(i) $f (x) = 2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$

(ii) $g (x) = x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1$

Q. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i)

2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2

(ii)

x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1

Q. Verify that the numbers given alongside that cubic polynomials below are its zeroes . Also verify the relationship between the zeroes and coefficient in each case.

2 x^{3} + x^{2} - 5 x + 2, (\frac{1}{2}, 1 . - 2)

Q. Verify that the numbers given along side of the cubic polynomials below are their zero.
(i)

f (x) = 2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2

(ii)

g (x) = x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1

Q. Verity that 3, −2, 1 are the zeros of the cubic polynomial p(x) = x³ − 2x² − 5x + 6 and verify the relation between its zeros and coefficients.

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Given the zeroes of a cubic polynomial f(x)=2x3+x2−5x+2;12,1,−2 Verify the relation between its zeros and coefficients.

Therefore, α=12β=1,γ=−2 now, Sum of roots = Products of roots = Sum of products of two consecutive roots = LHS=RHS Hence Verified

Given the zeroes of a cubic polynomial $f (x) = 2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$ Verify the relation between its zeros and coefficients.