Constructing a Polynomial with Specific Zeros

Understanding how to construct a polynomial with specific zeros is a fundamental skill in algebra. In this article, we will explore how to create a polynomial with a double zero at x 4 and a simple zero at x -3. We will walk through the steps and provide examples to illustrate the process.

Step-by-Step Construction

To construct a polynomial with a double zero at x 4 and a simple zero at x -3, we start by using the fact that each zero contributes a factor to the polynomial.

Double Zero at x 4

A double zero at x 4 means that the factor (x - 4)^2 must be included in the polynomial. This is because a double zero indicates that the factor appears twice.

Simple Zero at x -3

A simple zero at x -3 means that the factor (x 3) must be included in the polynomial. This is because a simple zero indicates that the factor appears once.

Combining the Factors

Combining these factors, we can express the polynomial as:

P(x) (x - 4)^2 (x 3)

Expanding the Polynomial

Next, we expand this polynomial to find the explicit form of P(x).

Step 1: Expand (x - 4)^2

We can expand (x - 4)^2 using the binomial theorem or simply by multiplying it out:

(x - 4)^2 x^2 - 8x 16

Step 2: Multiply the Result by (x 3)

Now, we multiply the result of the first step by (x 3) using the distributive property:

(x^2 - 8x 16)(x 3) x^3 3x^2 - 8x^2 - 24x 16x 48

Combining like terms, we get:

x^3 - 5x^2 - 8x 48

Final Form of the Polynomial

Thus, the polynomial with a double zero at x 4 and a simple zero at x -3 is:

P(x) x^3 - 5x^2 - 8x 48

Understanding the Form of the Polynomial

The above construction ensures that the polynomial has the required zeros. Adding linear factors with nonreal roots would not change the real roots of the polynomial, but could introduce complex roots, which is beyond the scope of this problem.

Examples of Polynomial Equations with the Same Roots

There are an infinite number of polynomials with the same zeros but different leading coefficients. For instance:

7(x - 4)^2 (x 3)2(x - 4)^2 (x 3)(x - 5)(x - 4)^2 (x 3)(x 1)

All these polynomials have the required double zero at x 4 and a simple zero at x -3.

Conclusion

By understanding how to construct polynomials with specific zeros, we can effectively model various mathematical and real-world scenarios. This article provides a systematic approach to creating such polynomials and demonstrates the flexibility in their construction.