Problem 31 - Precalculus Mathematics for Calculus

Question

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{4}-3 x^{3}+2 x^{2}$$

Step-by-Step Solution

Answer

The polynomial is factored as \(x^2(x - 2)(x - 1)\) with zeros at \(x = 0, 1, 2\).

Step 1: Factor out common terms

First, look for common terms in all the terms of the polynomial. In this case, we can factor out an \(x^2\), which gives us: \[ P(x) = x^2(x^2 - 3x + 2) \]

Step 2: Factor the quadratic expression

Next, we need to factor the quadratic expression \(x^2 - 3x + 2\). We look for two numbers that multiply to 2 (the constant term) and add up to -3 (the coefficient of \(x\)). These numbers are -2 and -1. Therefore, the quadratic can be factored as: \[ x^2 - 3x + 2 = (x - 2)(x - 1) \]

Step 3: Write the complete factored form

Now that we have factored the quadratic, we can write the complete factorization of the polynomial: \[ P(x) = x^2(x - 2)(x - 1) \]

Step 4: Determine the zeros from the factored form

To find the zeros of the polynomial, set each factor equal to zero and solve for \(x\): 1. \(x^2 = 0\) gives \(x = 0\) (this is a double root since \(x^2\))2. \(x - 2 = 0\) gives \(x = 2\)3. \(x - 1 = 0\) gives \(x = 1\)Thus, the zeros are \(x = 0, 1, 2\).

Step 5: Sketch the graph

The factored form \(P(x) = x^2(x - 2)(x - 1)\) tells us the zeros are \(x = 0, 1, 2\). Since \(x^2\) has multiplicity 2, the graph will touch the x-axis at \(x = 0\) and "bounce off", while it will pass through at \(x = 1\) and \(x = 2\). The graph is a quartic polynomial which is symmetric about the y-axis because the leading term is positive, and it will open upwards.

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions that involve sums and products of variables raised to whole number powers. They are characterized by their degree, which is determined by the highest power of the variable in the expression. In the exercise we have, the polynomial function is

• Quartic, meaning it has a degree of 4 because the highest power of the variable is 4.
• It is expressed as, \( P(x) = x^4 - 3x^3 + 2x^2 \).

Polynomial functions can be simple or complex, involving multiple terms. The behavior of the graph of a polynomial is closely related to its degree and leading coefficient.
In our example, the leading coefficient is positive, suggesting that for very large or small values of \(x\), the function will increase or decrease toward plus or minus infinity, respectively.

Breaking Down Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically expressed in the form \( ax^2 + bx + c \). In our exercise, while the original function is quartic, the factored expression includes a quadratic part:

• \( x^2 - 3x + 2 \)

To factor this expression, we need to find two numbers that multiply to the constant term (2) and add up to the linear coefficient (-3).
The numbers -2 and -1 fit this requirement, allowing us to write the quadratic as

• \( (x - 2)(x - 1) \).

Factoring quadratics is a cornerstone skill in solving polynomials because it simplifies the process of finding roots or zeros, which are the points where the graph intersects the x-axis.

Essentials of Graph Sketching

Drawing a graph of a polynomial function involves understanding its factored form, which provides the zeros or roots. The zeros are critical points where the graph of the function touches or crosses the x-axis. In our case, the factored form is

• \( x^2(x - 2)(x - 1) \)

This reveals that our polynomial has zeros at \(x = 0\), a double root, and \(x = 1\) and \(x = 2\), both single roots.

• The double root at \(x = 0\) means the graph will touch the x-axis and bounce back, not crossing it.
• Single roots at \(x = 1 \) and \(x = 2\), means the graph crosses the x-axis at these points.

Since the polynomial is quartic with a positive leading term, the ends of the graph will rise upwards on both sides. Sketching the graph, you'll see a curve that bounces off at \(x = 0\) and increases to cross the axis at \(x = 1\) and \(x = 2\). Understanding these aspects of graph behavior helps in predicting the shape and direction of the graph effectively.