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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: (i) 2x3+x2-5x+2; -1/2, 1, -2 (ii) x3-4x2+5x-2 ;2, 1, 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:

(i) 2x3+x2-5x+2; -1/2, 1, -2 (ii) x3-4x2+5x-2 ;2, 1, 1

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 8741 Points
3 months ago
Solution: Given, p(x) = 2x3+x2-5x+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 2x3+x2-5x+2. Now, comparing the given polynomial with general expression, we get; ∴ ax3+bx2+cx+d = 2x3+x2-5x+2 a=2, b=1, c= -5 and d = 2 As we know, if α, β, γ are the zeroes of the cubic polynomial ax3+bx2+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. (ii) x3-4x2+5x-2 ;2, 1, 1 Solution: Given, p(x) = x3-4x2+5x-2 And zeroes for p(x) are 2,1,1. ∴ p(2)= 23-4(2)2+5(2)-2 = 0 p(1) = 13-(4×12 )+(5×1)-2 = 0 Hence proved, 2, 1, 1 are the zeroes of x3-4x2+5x-2 Now, comparing the given polynomial with general expression, we get; ∴ ax3+bx2+cx+d = x3-4x2+5x-2 a = 1, b = -4, c = 5 and d = -2 As we know, if α, β, γ are the zeroes of the cubic polynomial ax3+bx2+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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