Problem 47 - College Algebra

Question

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one. Factor is \(x^{2}+x+1\)

Step-by-Step Solution

Answer

Factor the polynomial as \((x^2 + x + 1)(x - r)\), where \(r\) is found from the graph.

Step 1: Understand the Problem

We are given a third-degree polynomial in graphical form, but instead of the exact polynomial, we are provided with one factor: \(x^2 + x + 1\). Our task is to write the polynomial in its factored form, given that the leading coefficient is 1.

Step 2: Recognize the Structure of a Third-degree Polynomial

A third-degree polynomial with a leading coefficient of 1 will have the general form of \(x^3 + ax^2 + bx + c\). We are to factor this polynomial using the given factor \(x^2 + x + 1\).

Step 3: Determine the Remaining Factor

Since the polynomial is of third degree, and one of its factors is of second degree, the remaining factor must be of first degree. Let's call this factor \(x - r\). Thus, the polynomial can be expressed as the product \((x^2 + x + 1)(x - r)\).

Step 4: Expand the Expression to Check Consistency

Expand \((x^2 + x + 1)(x - r)\) to ensure it results in a third-degree polynomial. We perform the multiplication:\[x^3 - rx^2 + x^2 - rx + x - r = x^3 + (1 - r)x^2 + (1 - r)x - r\]This expression aligns with the general form \(x^3 + ax^2 + bx + c\), where we need to find \(r\) using given graph or conditions.

Step 5: Solve for the Missing Factor Zero

To find \(r\), we need a point or intercept from the graph. Usually, for polynomial factorization, this information is derived from visible x-intercepts or behavior indicated on the graph. Typically, the intercepts/known roots are used here to solve for \(r\).

Step 6: Verify the Result

Assume you derived \(r\) from the graph: substitute back and verify all factors are correct by re-expanding to the original polynomial. Re-evaluate against graph if necessary to ensure consistency.

Key Concepts

Third-Degree Polynomial

A third-degree polynomial is a polynomial with the highest degree of three. This means the largest exponent of the variable (usually \( x \)) is three. This kind of polynomial is also called a cubic polynomial. Cubic polynomials have a variety of forms, but their general structure can be captured as \( ax^3 + bx^2 + cx + d \). Here:

• \( a \) is the leading coefficient.
• \( b, c, \) and \( d \) are coefficients of the lower degree terms.

Because their degree is odd, third-degree polynomials behave differently than even-degree polynomials—they will always have at least one real root and the graph will always extend from one quadrant to another, crossing the x-axis at least once. Working with these polynomials requires understanding how to break down its structure and analyze its components, often leading to factorization tasks.

Factored Form

The factored form of a polynomial expresses the polynomial as a product of its factors. Factors are simpler polynomials that, when multiplied together, yield the original polynomial. Writing in factored form is useful for solving equations, as it can reveal the roots or solutions directly.

For a cubic polynomial, such as the one in the exercise, writing in factored form involves identifying each polynomial factor. In the given exercise, one factor is provided: \(x^2 + x + 1\). Our task involves deducing the remaining factors. Since it is a third-degree polynomial, a likely choice for the missing factor is a linear polynomial, represented here as \(x - r\). Once fully factored, the polynomial can be expressed as \((x^2 + x + 1)(x - r)\). Knowing the factored form helps determine x-intercepts or other critical points from the polynomial's graph.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a crucial role in determining the general shape and end behavior of the polynomial's graph. In the exercise, the leading coefficient is given as 1.

This is significant because:

• A positive leading coefficient (like 1) implies that as \(x\) approaches infinity, the polynomial's value also goes to infinity.
• A negative leading coefficient would mean the opposite end behavior.

The leading coefficient of 1 simplifies calculations greatly, as it implies that the polynomial is monic (leading coefficient = 1), which often makes factorizations and expansions more straightforward. Understanding whether your polynomial is monic can guide you in determining appropriate methods for factorization and solution.

Polynomial Graph

Graphing a polynomial gives a visual representation of its roots and behavior. For a third-degree polynomial, the graph will typically have the following characteristics:

• At most three roots or x-intercepts.
• Turns or stationary points (maximum or minimum).
• End behavior dictated by the leading coefficient: rising to the right and falling to the left if positive; the reverse if negative.

The exercise involves using a polynomial graph to aid in determining missing factors for the polynomial. The graph can hint at the correct value of \(r\) in the factor \((x - r)\). You would look at any visible x-intercepts, as they give clues about possible roots of the polynomial, each potentially suggesting a linear factor. Reading and interpreting these graphs are key skills that allow comprehensive understanding of polynomial behavior.