Exploring the Forms of Polynomial p(x) with Specific Roots and Conditions

In the realm of algebra, polynomials are fundamental objects of study. Over the course of this article, we will delve into a specific type of polynomial, p(x), which poses a unique challenge: it has roots at -1 and 2, and it is equal to 1 at x 0. This problem will require us to explore the properties of polynomials and understand the implications of the given conditions.

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, p(x) a_nx^n a_{n-1}x^{n-1} ... a_1x a_0, where a_n, a_{n-1}, ..., a_1, a_0 are constants and n is a non-negative integer.

Conditions and Analysis

Our task is to find a polynomial p(x) that satisfies two conditions:

It has roots at -1 and 2. The polynomial equals 1 when x 0.

Let's analyze these conditions one by one.

Roots of the Polynomial

A polynomial p(x) having roots at -1 and 2 can be expressed in the form:

Equation 1: ( p(x) c(x 1)(x - 2) )

where c is a constant. This form ensures that the polynomial will be zero at x -1 and x 2.

Condition at x 0

Given that p(0) 1, we substitute x 0 into the equation:

Equation 2: ( p(0) c(0 1)(0 - 2) 1 )

This simplifies to:

Equation 3: ( -2c 1 )

Solving for c, we get:

Equation 4: ( c -frac{1}{2} )

Form of the Polynomial

Substituting ( c -frac{1}{2} ) back into the general form, we get:

Equation 5: ( p(x) -frac{1}{2}(x 1)(x - 2) )

Expanding the polynomial, we get:

Equation 6: ( p(x) -frac{1}{2}(x^2 - x - 2) -frac{1}{2}x^2 frac{1}{2}x 1 )

This is the unique form of the polynomial that satisfies both conditions.

Conclusion

Through this analysis, we have found that the only possible form of the polynomial p(x) with roots at -1 and 2 and equal to 1 at x 0 is:

Equation 7: ( p(x) -frac{1}{2}x^2 frac{1}{2}x 1 )

This result showcases the intricate relationship between the roots of a polynomial and its form, emphasizing the importance of algebraic manipulation in understanding mathematical functions.

Key Takeaways

A polynomial with roots at -1 and 2 can be written as ( c(x 1)(x - 2) ). For p(0) 1, we determine that ( c -frac{1}{2} ). The unique form of the polynomial is ( p(x) -frac{1}{2}x^2 frac{1}{2}x 1 ).

Further Exploration

This problem can be extended to explore other polynomial behaviors, such as higher degrees or additional constraints. For instance, if we were to add another root or modify the constant value, how would this affect the polynomial's form? These questions can provide deeper insights into the nature of polynomials and their properties.