Problem 20 - Precalculus: Functions and Graphs

Question

Find a polynomial \(f(x)\) with leading coefficient 1 and having the given degree and zeros. degree \(4 ; \quad\) zeros \(-3,0,1,5\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = x^4 - 3x^3 - 13x^2 + 15x \).

Step 1: Write Down the Polynomial Formula

When given the zeros of a polynomial and the degree, we use the fact that a polynomial can be expressed in the form of its factors. Each zero corresponds to a factor of the form \((x - r)\), where \(r\) is a zero of the polynomial. A polynomial of degree \(4\) with zeros \(-3, 0, 1,\) and \(5\) is expressed as:\[f(x) = (x + 3)(x - 0)(x - 1)(x - 5)\]

Step 2: Simplify the Polynomial

Simplify the expression for \(f(x)\). Start by rewriting the factor \((x - 0)\) as \(x\):\[f(x) = (x + 3)x(x - 1)(x - 5)\]

Step 3: Expand the Polynomial

Expand the product step by step by first multiplying the simpler terms and then combining them:- First, expand \((x + 3)x:\) \[(x + 3)x = x^2 + 3x\]- Then, expand \((x - 1)(x - 5):\) \[(x - 1)(x - 5) = x^2 - 5x - 1x + 5 = x^2 - 6x + 5\]- Now, combine these results: \[f(x) = (x^2 + 3x)(x^2 - 6x + 5)\]

Step 4: Final Expansion

Expand \((x^2 + 3x)(x^2 - 6x + 5)\) using the distributive property (FOIL):- Multiply \(x^2\) with each term in \(x^2 - 6x + 5\) \[x^2(x^2 - 6x + 5) = x^4 - 6x^3 + 5x^2\]- Multiply \(3x\) with each term in \(x^2 - 6x + 5\) \[3x(x^2 - 6x + 5) = 3x^3 - 18x^2 + 15x\]- Sum these expressions: \[f(x) = x^4 - 6x^3 + 5x^2 + 3x^3 - 18x^2 + 15x\]- Simplify by combining like terms: \[f(x) = x^4 - 3x^3 - 13x^2 + 15x\]

Step 5: Confirm Leading Coefficient

Ensure the leading coefficient is \(1\), which it is, confirming the polynomial is in the correct form.

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial, often called roots, are the values of the variable that make the polynomial equal to zero. In simpler terms, these are the values you find for which the entire polynomial evaluates to zero. For a polynomial function like our example, those are

• \(-3\)
• 0
• 1
• 5

Each zero corresponds to a factor of \((x - r)\), where \(r\) is the zero. Therefore, if you have a zero at \(-3\) , the factor is \((x + 3)\), because solving \(x + 3 = 0\) gives \(x = -3\). Hence, zeros are crucial in constructing polynomial equations from their factors.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable within the polynomial. It tells you the maximum number of roots the polynomial can have, including both real and complex roots. In our polynomial, the degree is \(4\), because the polynomial is expressed in the form \(x^4 - 3x^3 - 13x^2 + 15x\), where the highest power of \(x\) is \(4\).
The degree gives crucial information about the polynomial's graph and behavior. For example, it tells us that there are potentially four zeros or roots, a conclusion supported by the specific zeros given in our problem.

Expanding Polynomials

Expanding a polynomial involves multiplying its factors, often expressed in the form of binomials, to transform it into a standard polynomial form. This usually means taking expressions like \((x + 3)(x - 1)(x + 5)\) and using methods like the distributive property (or FOIL method for binomials) to open up all the brackets and gather like terms.
For our polynomial:

• First, multiply the simpler terms like \((x + 3)x = x^2 + 3x\).
• Then, \((x - 1)(x - 5)\) becomes \(x^2 - 6x + 5\).
• Next, combine these results:\((x^2 + 3x)(x^2 - 6x + 5)\).

Repeat these steps until every term is multiplied out and all similar terms are combined. Expanding changes factored polynomials into their simpler equivalent form, ready for analysis or graphing.

Leading Coefficient

The leading coefficient of a polynomial is the number in front of the term with the highest degree after expanding and simplifying all polynomial terms. For example, in the polynomial \(f(x) = x^4 - 3x^3 - 13x^2 + 15x\), the leading coefficient is \(1\)because the coefficient in front of \(x^4\), the term with the highest power, is \(1\). A leading coefficient of \(1\) indicates that the polynomial is monic.
The leading coefficient is also significant because it affects the end behavior of the polynomial's graph, specifically, how steeply the graph rises or falls as you move away from the origin toward positive or negative infinity.