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+ 5x2; 2, 1, 1
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}+ 5x2; 2, 1, 1
Given, p(x) = x^{3}4x^{2}+5x2
And zeroes for p(x) are 2,1,1.
∴ p(2)= 2^{3}4(2)^{2}+5(2)2 = 0
p(1) = 1^{3}(4×1^{2 })+(5×1)2 = 0
Hence proved, 2, 1, 1 are the zeroes of x^{3}4x^{2}+5x2
Now, comparing the given polynomial with general expression, we get;
∴ ax^{3}+bx^{2}+cx+d = x^{3}4x^{2}+5x2
a = 1, b = 4, 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 β γ = – d/a.
Therefore, putting the values of zeroes of the polynomial,
α +β+γ = 2+1+1 = 4 = (4)/1 = –b/a
αβ+βγ+γα = 2×1+1×1+1×2 = 5
= 5/1= c/a
αβγ = 2×1×1 = 2 = (2)/1 = d/a
Hence, the relationship between the zeroes and the coefficients are satisfied.

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