Problem 53 - Algebra for College Students

Question

We can find the zeros of a polynomial function by solving a polynomial equation. We can also work backward to find a polynomial function that has given zeros. a) Write a first-degree polynomial function whose zero is \(-2\) b) Write a second-degree polynomial function whose zeros are 5 and \(-5\) c) Write a third-degree polynomial function whose zeros are \(1,-3,\) and 4 d) Is there a polynomial function with any given number of zeros? What is its degree?

Step-by-Step Solution

Answer

a) x + 2 b) x^2 - 25 c) x^3 - 2x^2 - 11x +12 d) Yes, its degree is equal to the number of zeros.

Step 1: Identify the Zero for First-Degree Polynomial

The problem states that the zero for the first-degree polynomial is (-2).

Step 2: Form the First-Degree Polynomial

Use the fact that the zero of a polynomial can be used to form its factor. Since the zero is -2, the factor form is (x + 2). Therefore, the polynomial is (x + 2).

Step 3: Identify the Zeros for the Second-Degree Polynomial

Given zeros are 5 and (-5).

Step 4: Form the Second-Degree Polynomial

Since (x - a) and (x - b) are factors when a and b are zeros, the second-degree polynomial is (x - 5)(x + 5). Expanding this gives (x^2 - 25).

Step 5: Identify the Zeros for the Third-Degree Polynomial

Given zeros are 1, -3, and 4.

Step 6: Form the Third-Degree Polynomial

Using the zeros, the factors are (x - 1), (x + 3), and (x - 4). Therefore, the polynomial is (x - 1)(x + 3)(x - 4). Expanding this product gives (x^3 - 2x^2 - 11x + 12).

Step 7: Determine the Relationship Between Number of Zeros and Degree

A polynomial with n zeros is an nth-degree polynomial. Thus, a polynomial with any given number of zeros will have its degree equal to the number of zeros it has.

Key Concepts

zeros of polynomials

In a polynomial function, the 'zeros' are the values of \( x \) that make the polynomial equal to zero. For example, if \( f(x) = (x+2) \), the zero of this polynomial is \( x = -2 \), since substituting \( -2 \) for \( x \) in the polynomial gives zero. Zeros are also referred to as the 'roots' of the polynomial. To find the zeros, you can set the polynomial equal to zero and solve for \( x \). Steps to find zeros:

• Set the polynomial equal to zero: \( f(x) = 0 \).
• Factorize the polynomial, if possible.
• Solve the resulting equations for \( x \).

Practicing finding zeros helps in graphing and understanding polynomial behavior.

degree of polynomial

The degree of a polynomial is the highest power of \( x \) in the polynomial. For example, in \( f(x) = x^3 - 2x^2 - 11x + 12 \), the highest power of \( x \) is 3, so it is a third-degree polynomial. Knowing the degree helps in understanding the polynomial's complexity and behavior. Characteristics based on degree:

• A first-degree polynomial is a linear function.
• A second-degree polynomial is a quadratic function.
• A third-degree polynomial is a cubic function.

Higher-degree polynomials have more complex graphs, with more turning points and zeros.

factoring polynomials

Factoring polynomials means expressing the polynomial as a product of simpler polynomials, called factors. For example, factoring \( x^2 - 25 \) gives \( (x - 5)(x + 5) \). It's useful for solving polynomial equations and finding zeros. Steps to factor polynomials:

• Look for a common factor in all terms.
• Use factoring formulas, such as the difference of squares \( a^2 - b^2 = (a - b)(a + b) \).
• Decompose the polynomial into linear or simpler quadratic factors.

Factoring is crucial for simplifying polynomials and solving polynomial equations.

polynomial equations

A polynomial equation is an equation of the form \( P(x) = 0 \, where \ P(x) \) is a polynomial function. Solving such equations means finding the zeros of the polynomial. For example, solving \( (x - 1)(x + 3)(x - 4) = 0 \) involves finding the values of \ x \ for which this product is zero. Steps to solve polynomial equations:

• Factor the polynomial completely.
• Set each factor equal to zero.
• Solve each resulting simple equation for \ x \.

Polynomial equations often arise in various fields, making this skill widely applicable.