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Verify that `3, -1`and `-1/3`are the zeros of the cubicpolynomial `p(x)=3x^3-5x^2-11 x-3`andthen verify the relationship between the zeros and its coefficients. |
Answer» The given polynomial is `p(x) = 3x^(3) - 5x^(2) = 11x - 3.` `:. P(3) = {3 xx 3^(3) - 5 xx 3^(2) -11 xx 3- 3} = (81-45 - 33-3) = 0,` `p(-1) ={3 xx (-1)^(3)-5xx(-1)^(2)-11 xx (-1)-3}` ` = (-3-5+11-3) = 0`, ` and p((-1)/3) = {3xx((-1)/3)^(3) - 5 xx ((-1)/3)^(2) - 11 xx ((-1)/3) - 3}` ` = {3 xx ((-1)/27) - 5 xx 1/9 + 11/3 - 3} = ((-1)/9-5/9+11/3-3)` ` = ((-1-5+33-27))/9 = 0.` ` :. 3,-1 and (-1)/3` are the zeros of p(x). Let ` alpha =3, beta=-1 and gamma = 1/3.` Then, `(alpha+beta+gamma) = (3-1-1/3) = 5/3 = (-("coefficient of " x^(2)))/(("coefficient of " x^(3))).,` `(alpha beta+beta gamma+gamma alpha) = (-3+1/3-1) = (-11)/3 = (("coefficient of x"))/(("coefficient of " x^(3)))., ` `alpha beta gamma ={3xx(-1)xx((-1)/3)} = 1=3/3 = (-("constant term"))/(("coefficient of " x^(3))).` |
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