Find the zeroes of the following polynomial and verify the rwlationship between the zeroes and coefficients. (i)4s^2-4s 1?

Question

Answer

Answer

Given,

Quadratic eq. = 4x² - 4x + 1

Here,

Coefficient of x² ( a ) = 4

Coefficient of x ( b ) = -4

Constant term ( c ) = 1

Now,

=> 4x² - 4x + 1 = 0

=> ( 2x )² - 2 × 2x × 1 + ( 1 )² = 0

Using identity :

=> ( a² - 2ab + b² ) = ( a - b )²
=> ( 2x - 1 )² = 0

=> ( 2x - 1 ) ( 2x - 1 ) = 0 ( continued further )

=> ( 2x - 1 ) = 0 ÷ ( 2x - 1 )

=> ( 2x - 1 ) = 0

=> x = 1/2

=> ( 2x - 1 ) = 0 ÷ ( 2x - 1 )

=> ( 2x - 1 ) = 0

=> 2x = 1

=> x = 1/2

Hence, zeroes are ( 1/2 ) and ( 1/2 ).

Now,

=> Sum of zeroes = -b/a

=> ( 1/2 ) + ( 1/2 ) = -( -4 ) ÷ 4

=> ( 1 + 1 )/2 = 4 ÷ 4

=> 2÷2 = 1

=> 1 = 1

And,

=> Product of zeroes = c/a

=> ( 1/2 ) × ( 1/2 ) = 1/4

1/4=1/4 ,

Hence, verified

Answer

4s^2 - 4s + 1 = 0
4s^2 - 2s - 2s + 1 = 0
2s ( 2s - 1) - 1 ( 2s - 1) = 0
(2s - 1) ( 2s - 1)
2s - 1 = 0
2s = 1
s = 1/2
sum of zeroes = - b / a = - ( -4 ) / 4 = 4 / 4 = 1
product of zeroes = c / a = 1 / 4

Answer

Answer

Actually your question is slightly incorrect

question is 4s²-4s+1

answer

let 4s²-4s+1=0
4s²-2s-2s+1=0
take common things
2s(2s-1)-1(2s-1)=0
(2s-1) (2s-1) =0
(2s-1) =0 and another (2s-1)=0

value of S = 2s-1=0
S= 2s=1
S=1/2

therefore zeroes of that polynomial is 1/2,1/2

then sum of the zeroes is 1/2+1/2=1(2/2)

then product of zeroes is 1/2(1/2)= 1/4

constant term/coefficient of S

Answer

Zeroes and Relationship between Zeroes and Coefficients of a Polynomial

Polynomial:

4s^2-4s+1

Finding Zeroes:

Let's use the quadratic formula to find the zeroes:

s = (-b ± sqrt(b^2-4ac))/2a

Here, a = 4, b = -4, and c = 1

Substituting the values, we get:

s = (-(-4) ± sqrt((-4)^2-4(4)(1)))/2(4)

s = (4 ± sqrt(16-16))/8

s = (4 ± 0)/8

So, we get two zeroes:

s = 1/2

Relationship between Zeroes and Coefficients:

Let's write the polynomial in factored form:

4s^2-4s+1 = (2s-1)(2s-1)

From the factored form, we can see that the polynomial has only one root, which is 1/2. This root has a multiplicity of 2.

The relationship between the zeroes and coefficients of a polynomial is given by Vieta's formulas:

The sum of the zeroes is equal to the coefficient of the second term divided by the coefficient of the leading term.

The product of the zeroes is equal to the constant term divided by the coefficient of the leading term.

Here, the sum of the zeroes is:

1/2 + 1/2 = 1

The coefficient of the second term is -4 and the coefficient of the leading term is 4, so:

sum of zeroes = -(-4)/4 = 1

The product of the zeroes is:

1/2 × 1/2 = 1/4

The constant term is 1 and the coefficient of the leading term is 4, so:

product of zeroes = 1/4

Thus, we can see that the relationship between the zeroes and coefficients of the given polynomial satisfies Vieta's formulas.

Answer

Answer

Answer:

Given quadratic polynomial is

4s2 − 4s + 1

To find the zeros of the quadratic polynomial we consider

4s2 − 4s + 1 = 0

4s2 − 2s – 2s + 1 = 0

2s(2s – 1) -1(2s – 1) = 0

(2s – 1)(2s – 1) = 0

2s – 1 = 0 and 2s – 1 = 0

2s = 1 and 2s = 1

s = 1/2 and s = 1/2

∴ The zeroes of the polynomial = 1/2 and 1/2

Sum of the zeroes = -(coefficient of s)/(coefficient of s2)

Sum of the zeroes = -(-4)/4 = 1

Let’s find the sum of the roots = 1/2 + 1/2 = 1

Product of the zeros = Constant term / Coefficient of s2

Product of the zeros =1 / 4

Let’s find the products of the roots = 1/2 × 1/2 = 1/4

relationship verified

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