Problem 68 - Algebra 2

Question

Write a polynomial function in standard form with the given zeros. 1, multiplicity 4

Step-by-Step Solution

Answer

The polynomial function in standard form with the given zero 1 and multiplicity 4 is \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \).

Step 1: Understand the Zero and its Multiplicity

The given zero is 1, and it has a multiplicity of 4. This means that the factor corresponding to this zero will appear four times in the polynomial.

Step 2: Write Down the Factor

Since the zero of the polynomial is 1, the factor must be in the form \(x - a\) where \(a\) is the zero. Here, \(a = 1\), so the factor is \(x - 1\).

Step 3: Account for the Multiplicity

With a multiplicity of 4, the factor \(x - 1\) will be repeated 4 times. Therefore, the polynomial will have \(x - 1\) to the fourth power: \( (x - 1)^4 \).

Step 4: Expand the Factor

To write the polynomial in standard form, expand the factor: \( (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 \).

Step 5: Write the Polynomial in Standard Form

The polynomial in standard form, having the zero 1 with multiplicity 4, is \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \).

Key Concepts

Polynomial Zeros

In the world of polynomials, a 'zero' or a 'root' of a polynomial is a special value for which the polynomial gives an output of zero. In simpler terms, if you plug this value into the polynomial, the result is zero. It's like the point where the graph of the polynomial touches or crosses the x-axis.

For example, if a polynomial has a zero at 3, it means that when you replace the variable in the polynomial with 3, the whole expression equals zero. Mathematically, if you have a polynomial function such as \( f(x) = x^2 - 9 \) and you input 3, you get \( f(3) = 3^2 - 9 = 0 \), confirming that 3 is indeed a zero of the polynomial.

Multiplicity of Zeros

The 'multiplicity' of a zero refers to how many times a particular zero is repeated as a factor of the polynomial. It's closely associated with the behavior of the graph at that zero. A zero with a multiplicity of 1, known as a 'simple' zero, means the graph will cross the axis. A zero with an even multiplicity indicates that the graph will just touch the x-axis and turn back. A zero with an odd multiplicity greater than 1 will still cross the axis but with a flatter, more gradual approach.

Let's take the zero with a multiplicity of 4 from our exercise. The zero is 1, and because its multiplicity is an even number, the graph of our polynomial will touch the x-axis at x=1 and then bounce back, without actually crossing the axis.

Standard Form of a Polynomial

A polynomial is in 'standard form' when its terms are ordered by their degrees in descending order, starting from the highest degree to the constant term, and the leading coefficient (the coefficient of the term with the highest degree) should be positive. For example, the standard form of a quadratic polynomial is \( ax^2 + bx + c \), with the degree of \( x^2 \) being the highest.

When writing a polynomial in standard form, like we did with \( (x - 1)^4 \), we start by expanding it (which we will discuss next) and then rearrange the terms as needed to make sure they are in the correct order. This form is crucial for many types of analysis and solving, making it easier to read and understand the hierarchy of terms by degree.

Expanding Polynomials

To 'expand' a polynomial means to multiply out the factors of the expression to eliminate parentheses and combine like terms. This process is fundamental to simplifying expressions and is especially necessary when we deal with polynomials that have binomial factors raised to a power, as in our example with \( (x - 1)^4 \).

Expanding can be done through several methods such as the distributive property, FOIL (for two binomials), and for binomials raised to higher powers, techniques such as Pascal's Triangle or the Binomial Theorem can be utilized. After expanding \( (x - 1)^4 \) as demonstrated in the example, we rearrange the terms in descending order of their power to achieve the standard form of the polynomial.