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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: x ^ 3 - 4x ^ 2+ 5x - 2

Q1 (2)  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:

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

Answers (1)
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S Sayak

p(x) = x3 - 4x2 + 5x - 2

p(2) = 23 - 4 x 22 + 5 x 2 - 2

p(2) = 8 - 16 + 10 - 2

p(-2) = 0

p(1) = 13 - 4 x 12 + 5 x 1 - 2

p(1) = 1 - 4 + 5 - 2

p(1) = 0

Therefore the numbers given alongside the polynomial are its zeroes

Verification of relationship between the zeroes and the coefficients

Comparing the given polynomial with ax3 + bx2 + cx + d, we have

a = 1, b = -4, c = 5, d = -2

The roots are \alpha ,\beta \ and\ \gamma

\\\alpha=2\\ \beta =1\\ \gamma =1

\\\alpha+\beta +\gamma \\ =2+1+1\\ =4\\ =-\frac{-4}{1}\\=-\frac{b}{a}

Verified

\\\alpha\beta +\beta \gamma +\gamma \alpha \\ =2\times 1+1\times 1+1\times 2\\ =5\\ =\frac{5}{1}\\ =\frac{c}{a}

Verified

\\\alpha\beta\gamma \\=2\times 1\times 1\\=2 \\=-\frac{-2}{1}\\=-\frac{d}{a}

Verified

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