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) $2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$

(ii) $x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1$

Question

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)

2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2

(ii)

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

Filo · Accepted Answer

(i) 2x3+x2−5x+2;21​,1,−2p(x)=2x3+x2−5x+2          .... (1)Zeroes for this polynomial are 21​,1,−2Substitute the x=21​ in equation (1)p(21​)=2(21​)3+(21​)2−5(21​)+2=41​+41​+25​+2=0Substitute the x=1 in equation (1)p(1)=2×13+12−5×1+2=2+1−5+2=0Substitute the x=−2 in equation (1)p(−2)=2(−2)3+(−2)2−5(−2)+2=−16+4+10+2=0Therefore, 21​,1,−2 are the zeroes of the given polynomial.Comparing the given polynomial with ax3+bx2+cx+d we obtain,a=2,b=1,c=−5,d=2Let us assume α=21​, β=1, γ=−2Sum of the roots = α+β+γ=21​+1=2=2−1​=a−b​αβ+βγ+αγ=21​+1(−2)+21​(−2)=2−5​=ac​Product of the roots = αβγ=21​×x×(−2)=2−2​=ad​Therefore, the relationship between the zeroes and coefficient are verified.(ii) x3−4x2+5x−2;2,1,1p(x)=x3−4x2+5x−2      .... (1)Zeroes for this polynomial are 2,1,1Substitute x=2 in equation (1)p(2)=23−4×22+5×2−2=8−16+10−2=0Substitute x=1 in equation (1)p(1)=x3−4x2+5x−2=13−4(1)2+5(1)−2=1−4+5−2=0Therefore, 2,1,1 are the zeroes of the given polynomial.Comparing the given polynomial with ax3+bx2+cx+d we obtain,a=1,b=−4,c=5,d=−2Let us assume α=2, β=1, γ=1Sum of the roots = α+β+γ=2+1+1=4=−1−4​a−b​Multiplication of two zeroes taking two at a time=αβ+βγ+αγ=(2)(1)+(1)(1)+(2)(1)=5=15​=ac​Product of the roots = αβγ=2×1×1=2=−1−2​=ad​Therefore, the relationship between the zeroes and coefficient are verified.

Question Text	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) $2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$ (ii) $x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1$
Answer Type	Text solution:1
Upvotes	150

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) $2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$

(ii) $x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1$

Text solutionVerified

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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:(i) 2x3+x2−5x+2;21​,1,−2 (ii) x3−4x2+5x−2;2,1,1

Text solutionVerified

Practice questions from similar books

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) $2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$

(ii) $x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1$