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: 2x^3+x^25x+2; 1/2, 1, 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: 2x^{3}+x^{2}5x+2; 1/2, 1, 2
Given, p(x) = 2x^{3}+x^{2}5x+2
And zeroes for p(x) are = 1/2, 1, 2
∴ p(1/2) = 2(1/2)^{3}+(1/2)^{2}5(1/2)+2
= (1/4)+(1/4)(5/2)+2 = 0
p(1) = 2(1)^{3}+(1)^{2}5(1)+2 = 0
p(2) = 2(2)^{3}+(2)^{2}5(2)+2 = 0
Hence, proved 1/2, 1, 2 are the zeroes of 2x^{3}+x^{2}5x+2.
Now, comparing the given polynomial with general expression, we get;
∴ ax^{3}+bx^{2}+cx+d = 2x^{3}+x^{2}5x+2
a=2, b=1, c= 5 and d = 2
As we know, if α, β, γ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;
α +β+γ = –b/a
αβ+βγ+γα = c/a
α βγ = – d/a.
Therefore, putting the values of zeroes of the polynomial,
α+β+γ = ½+1+(2) = 1/2 = –b/a
αβ+βγ+γα = (1/2×1)+(1 ×2)+(2×1/2)
= 5/2 = c/a
α β γ = ½×1×(2) = 2/2 = d/a
Hence, the relationship between the zeroes and the coefficients are satisfied.

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