Find the Zeros of Polynomial p(x)=4x^2-4x+1 and Verify the Relationship between the Zeros and the Coefficients

Find the Zeros of Polynomial p(x)=4x²-4x+1 and Verify the Relationship between the Zeros and the Coefficients

Solution:

To find the zeros of the polynomial p(x) = 4x² – 4x + 1, we need to solve the equation p(x) = 0.

We can use the quadratic formula to find the solutions. The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions (zeros) are given by:

x = (-b ± √(b² – 4ac)) / (2a)

In our case, a = 4, b = -4, and c = 1. Substituting these values into the quadratic formula, we have:

x = (-(-4) ± √((-4)² – 4(4)(1))) / (2(4))

Simplifying further:

x = (4 ± √(16 – 16)) / 8 x = (4 ± √0) / 8 x = (4 ± 0) / 8

The discriminant (b² – 4ac) is 0, indicating that the quadratic equation has only one solution.

Simplifying further:

x = 4/8 = 1/2

Therefore, the zero of the polynomial p(x) = 4x² – 4x + 1 is x = 1/2.

To verify the relationship between the zeroes and the coefficients, we can use Vieta’s formulas. For a quadratic equation of the form ax² + bx + c = 0, the sum of the zeros is given by:

Sum of zeros = -b/a

In our case, a = 4 and b = -4. Substituting these values into the formula:

Sum of zeros = -(-4)/4 Sum of zeros = 4/4 Sum of zeros = 1

The sum of the zeros is 1, which matches the coefficient of the linear term (-4x).

Now the product of the zeros is given by:

Product of zeros = c/a

In our case, a = 4 and c = 1. Substituting these values into the formula:

Product of zeros = 1/4

The product of the zeros is 1/4, which matches the constant term (1) in the polynomial.

Therefore, we have verified the relationship between the zero (1/2) and the coefficients of the polynomial p(x) = 4x² – 4x + 1 using Vieta’s formulas.

Answer:

Zeros of Polynomial p(x)=4x²-4x+1 is x = 1/2

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