HOW TO FIND ZERO OF A POLYNOMIAL FUNCTION

To find zeroes of a polynomial, we have to equate the polynomial to zero and solve for the variable.

To check whether 'k' is a zero of the polynomial f(x), we have to substitute the value 'k' for 'x' in f(x). 

If f(k) = 0, then 'k' is a zero of the polynomial f(x).

Find the zeros of the following polynomials

Question 1 :

p(x)  =  4x-1

Solution :

Let p(x)  =  4x-1 

4x-1  =  0

4x  =  1

x  =  1/4

So, 1/4 is the zero of the given polynomial 4x-1.

Question 2 :

p(x)  =  3x+5

Solution :

Let p(x)  =  3x+5

3x+5  =  0

3x  =  -5

x  =  -5/3

So, -5/3 is the zero of the given polynomial 3x+5.

Question 3 :

p(x)  =  2x

Solution :

Let p(x)  =  2x

2x  =  0

x  =  0/2

x  =  0

So, 0 is the zero of the given polynomial 2x.

Question 4 :

p(x)  =  x+9

Solution :

Let p(x)  =  x+9

x+9  =  0

x  =  -9

So, -9 is the zero of the given polynomial x+9.

Verify whether the following are roots of the polynomial equations indicated against them.

Question 1 :

x2 - 5x +6  =  0      x  =  2, 3 

Solution :

Given that p(x) =  x² - 5x +6

p(2) =  22 5(2) + 6

=  4 -10 + 6

p(2)  =  0

Given that p(x)   =  x2 - 5x +6

p(3)  =  32-5(3)+6

=  9 -15 + 6

p(3) =  0

So, 2 and 3 are zeroes or roots of the polynomial p(x).

Question 2 :

x2 +4x+3  =  0 : x  = -1, 2

Solution :

Given that p(x)  =  x2 +4x+3

p(-1) =  (-1)2+ 4(-1) +3

=   1-4+3

p(-1)  =  0

p(2)  =  22+4(2)+3

=  4+8+3

p(2)  =  11 ≠ 0

So, -1 is a zero of the polynomial and 2 is not a zero.

Question 3 :

x3 -2x2-5x+6  =  0 : x  = 1, -2, 3

Solution :

Given that p(x)  =  x3 -2x2-5x+6

p(-1) =  x3 -2x2-5x+6

p(1) =  (1)3 -2(1)2-5(1) +6

=  1-2-5+6

p(-1)  = 0

p(-2) =  (-2)3 -2(-2)2-5(-2) +6

=  -8-8+10+6

=  -16+16

p(-2)  =  0

p(3) =  33 -2(3)2-5(3) +6

=  27-18-15+6

=  33-33

p(3)  =  0

So, -1, -2 and 3 are zeroes or roots of the given polynomial.

Question 4 :

x3-2x2-x+2  =  0:     x =  -1, 2

Solution :

Given that p(x)  =  x3-2x2-x+2

p(-1) =  x3-2x2-x+2

p(-1) =  (-1)3 -2(-1)2-(-1)+2

=  -1-2+1+2

=  -3+3

p(-1)  =  0

p(2) =  23 -2(2)2-2+2

=  8-2(4)-2+2

=  8-8-2+2

p(2)  =  0

So, -1 and 2 are zeroes of the polynomial.

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