Question

Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in in each case:

(i) $f\left(x\right)=2x^3+x^2-5x+2;\frac{1}{2},1,-2$

(ii) $g\left(x\right)=x^3-4x^2+5x-2;2,1,1$

Answer 1

$2x^3+x^2-5x+2$

$=2x^3-2x^2+3x^2-3x-2x+2$

$=2x^2(x-1)+3x(x-1)-2(x-1)$

$=(x-1)(2x^2+3x-2)$

$=(x-1)(2x^2+4x-x-2)$

$=(x-1)\left[2x\right(x+2)-(x+2\left)\right]$

$=(x-1)(x+2)(2x-1)$
Hence,
x = 1, -2 , 1/2 are the zeros of
given numbers are same as evaluate numbers . hence, verified .

now,
Sum of roots =


Products of roots =

Sum of products of two consecutive roots =

LHS=RHS

Hence Verified

$x^3-4x^2+5x-2$

$=x^3-x^2-3x^2+3x+2x-2$

$=x^2(x-1)-3x(x-1)+2(x-1)$

$=(x-1)(x^2-3x+2)$

$=(x-1)(x-1)(x-2)$

Hence, 2, 1,1 are the zeros of
Given numbers are same as evaluate numbers so, verified .

Now,
Sum of roots :
=> -(-4)= 2 + 1 + 1
=> 4 = 4

Product of roots :
=> -(-2) = 2 × 1 × 1
=> 2 = 2

Sum of products of two consecutive roots :
=> 5 = 2 × 1 + 1 × 1 + 1 × 2
=> 5 = 5
Hence, verified