What is a Quadratic Polynomial Given Its Zeroes?

When given the zeroes of a quadratic polynomial, one can easily derive the polynomial expression. This article delves into the process of finding a quadratic polynomial given its zeroes, specifically focusing on the zeroes 5√2 and 5 - √2. We will explore the step-by-step derivation and the mathematical principles involved.

Defining the Polynomial Expression

Given the zeroes ( r_1 5√2 ) and ( r_2 5 - √2 ), the quadratic polynomial can be expressed as:

P(x) a(x - r_1)(x - r_2)

where ( a ) is a non-zero constant, typically 1 for simplicity. Thus, the polynomial becomes:

P(x) (x - 5√2)(x - (5 - √2))

Deriving the Polynomial Expression

We start by expanding the polynomial given its zeroes:

Step 1: Expand the Expression

P(x) (x - 5√2)(x - 5 √2)

Using the distributive method (FOIL) to expand:

P(x) x(x - 5 √2) - 5√2(x - 5 √2)

Step 2: Simplify the Expression

Let's distribute the terms:

P(x) x^2 - 5x x√2 - 5√2x 25 - 5√2√2

Simplify the expression:

P(x) x^2 - 5x x√2 - 5x√2 25 - 2

Merge and simplify:

P(x) x^2 - 5x x√2 - 5x√2 23

Combine the like terms:

P(x) x^2 - 1 23

Final Polynomial Expression

Thus, the quadratic polynomial whose zeroes are 5√2 and 5 - √2 is:

P(x) x^2 - 1 23

Summary of Key Points

The zeroes of a quadratic polynomial are 5√2 and 5 - √2. The polynomial can be expressed as (x - 5√2)(x - 5 √2). Using the distributive method (FOIL), the polynomial simplifies to x^2 - 1 23. This polynomial can also be derived by summing and multiplying the zeroes.

Mathematical Formulae

Sum of the roots: ( alpha beta 10 ) Product of the roots: ( alpha cdot beta 23 )

For further understanding, you can always refer to the following key concepts:

Quadratic Polynomial: A polynomial of degree 2 given by P(x) ax^2 bx c. Zeroes: The values of x for which P(x) equals zero.

Remember, the process of deriving a quadratic polynomial from its zeroes is straightforward and can be applied to similar problems.