Question

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:
@ 2x'+x2–5x+2; 3.1.-2
(i) -4r + 5x - 2; 2,1,1​

Answer 1

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 2x3+x2−5x+2;21,1,−2

p(x)=2x3+x2−5x+2          .... (1)

Zeroes for this polynomial are 21,1,−2

Substitute the x=21 in equation (1)

p(21)=2(21)3+(21)2−5(21)+2

=41+41+25+2

=0

Substitute the x=1 in equation (1)

p(1)=2×13+12−5×1+2

=2+1−5+2=0

Substitute the x=−2 in equation (1)

p(−2)=2(−2)3+(−2)2−5(−2)+2

=−16+4+10+2=0

Therefore, 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=2

Let us assume α=21, β=1, γ=−2

Sum 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;2,1,1

p(x)=x3−4x2+5x−2      .... (1)

Zeroes for this polynomial are 2,1,1

Substitute x=2 in equation (1)

p(2)=23−4×22+5×2−2

=8−16+10−2=0

Substitute x=1 in equation (1)

p(1)=x3−4x2+5x−2

=13−4(1)2+5(1)−2

=1−4+5−2=0

Therefore, 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=−2

Let us assume α=2, β=1, γ=1

Sum of the roots = α+β+γ=2+1+1=4=−1−4a−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.