Relationship of Zeros With Coefficients of Cubic Polynomial


 
 
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Relationship of Zeros With Coefficients of Cubic Polynomial

Relationship of zero with coefficients of cubic polynomial:

A cubic polynomial ax^3+bx^2+cx+d, aneq 0 has exactly three zeros. Let these be alpha ,beta ;and;gamma , then

alpha+beta+gamma= -frac{b}{a}                      [sum of the zeros]

beta gamma +gamma alpha +alpha beta =frac{c}{a}                  [sum of the product of zeros taken two at a time]

alphabetagamma =-frac{d}{a}                                   [product of zeros]

Example: Verify that 1/2, 1, -2 are zeros of 2x^3+x^2-5x+2. Also verify the relationship between the zeros and the coefficients.

Solution: Let f(x)=2x^3+x^2-5x+2

we have

f(frac{1}{2})=2(frac{1}{2})^3+(frac{1}{2})^2-5(frac{1}{2})+2

=frac{1}{2}+(frac{1}{2})^2-5(frac{1}{2})+2

f(1)=2+1-5+2=0

and    f(-2)=2(-2)^{3}+(-2)^{2}-5(-2)+2

                       =-16+4+10+2=0

Thus, frac{1}{2},1;and;-2  are zeros of f(x).

We have

           frac{1}{2}+1+(-2)=-frac{1}{2},

(1)(-2)+frac{1}{2}(-2)+1(-2)=-frac{5}{2}

           left ( frac{1}{2} right )(1)(-2)=-1

Note that in each case LHS = RHS.

 
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