Problem 6 - Algebra for College Students

Question

What are two ways to determine whether \(c\) is a zero of a polynomial?

Step-by-Step Solution

Answer

Evaluate \(P(c)\) to see if it equals zero or check if \(x - c\) is a factor of \(P(x)\).

Step 1: Use Polynomial Evaluation

To determine if a value, say \(c\), is a zero of a polynomial, substitute \(c\) into the polynomial. If the result is zero, then \(c\) is a zero of the polynomial. Mathematically, this means if \(P(c) = 0\), then \(c\) is a zero of the polynomial \(P(x)\).

Step 2: Factorization Method

Another method to determine if \(c\) is a zero of a polynomial is by factorization. If \(x - c\) is a factor of the polynomial \(P(x)\), then \(c\) is a zero of the polynomial. To check, factor the polynomial completely and see if \(x - c\) appears as one of the factors.

Key Concepts

Polynomial Evaluation

When you want to determine if a number, let's call it \(c\), is a zero of a polynomial, the first approach you can use is polynomial evaluation. This method involves substituting the value of \(c\) into the polynomial equation and solving it to see if the output is zero.
To put it simply, if you have a polynomial \(P(x)\), replace \(x\) with \(c\) and calculate the result. If the outcome is zero, then \(c\) is indeed a zero of the polynomial. The mathematical condition is expressed as \(P(c) = 0\).
For example, suppose we have a polynomial \(P(x) = x^2 - 5x + 6\). If we want to check if \(c = 2\) is a zero of this polynomial, we substitute 2 into the polynomial:
\[ P(2) = 2^2 - 5(2) + 6 \] Simplifying this, we get:
\[ P(2) = 4 - 10 + 6 = 0 \]
Since the result is zero, 2 is a zero of the polynomial \(P(x) = x^2 - 5x + 6\). This method is straightforward and works well for checking specific values. If you get a non-zero result, then \(c\) is not a zero of the polynomial.

Factorization

Another powerful method to determine if \(c\) is a zero of a polynomial is through factorization.
This approach involves breaking down the polynomial into factors. If one of the factors is \(x - c\), then \(c\) is a zero of the polynomial.
The idea is simple: if \(x - c\) can divide the polynomial without leaving a remainder, then \(c\) must be a zero. To find out, you need to factor the polynomial completely.
Let's look at an example. Consider the polynomial \(P(x) = x^2 - 5x + 6\). When we factor it, we get:
\[P(x) = (x - 2)(x - 3)\]
Here, we see the factors \(x - 2\) and \(x - 3\). These factors tell us that \(P(x)\) is zero when \(x = 2\) and when \(x = 3\). So, 2 and 3 are zeros of the polynomial \(P(x)\).
In summary, the factorization method helps us identify all the zeros of a polynomial by expressing it as a product of its factors. This method is especially useful when dealing with polynomials of higher degrees.

Zeros of Polynomials

Zeros of polynomials are the values of \(x\) that make the polynomial equal to zero.
Finding these zeros is critical because they provide insight into the polynomial's roots and its graphical behavior on a coordinate plane.
To determine the zeros of a polynomial, you can use methods such as polynomial evaluation and factorization as discussed earlier.
Polynomials can have multiple zeros, and they can be real or complex numbers.
- Real zeros are values that you can find on the real number line.
- Complex zeros include imaginary numbers, which are not located on the real number line.
For instance, in the polynomial \(P(x) = x^2 - 5x + 6\), the zeros are 2 and 3. These are real zeros. Similarly, a polynomial like \(P(x) = x^2 + 1\) has zeros that are not real, specifically \(i\) and \(-i\), where \(i = \sqrt{-1}\).
Understanding the zeros of a polynomial helps in graphing the equation and solving polynomial equations effectively. It also provides a deeper comprehension of the polynomial's characteristics and behavior.