Problem 7 - Precalculus Mathematics for Calculus

Question

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{3}-2 x^{2}+2 x$$

Step-by-Step Solution

Answer

The zeros are \(0, 1 + i, 1 - i\). The polynomial factors as \(x(x^2 - 2x + 2)\).

Step 1: Find the Common Factor

First, find the common factor of all terms in the polynomial \(P(x) = x^3 - 2x^2 + 2x\). Each term contains \(x\) as a factor. Thus, we can factor \(x\) out from the polynomial, obtaining: \(P(x) = x(x^2 - 2x + 2)\).

Step 2: Find the Zeros of the Factored Term

Focus on the quadratic equation \(x^2 - 2x + 2 = 0\) to find its zeros. We use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = -2, c = 2\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) of the quadratic:\( (-2)^2 - 4(1)(2) = 4 - 8 = -4\). This discriminant is negative, indicating that the roots are complex.

Step 4: Use the Quadratic Formula for Complex Roots

Substitute into the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{-4}}{2 \cdot 1} = \frac{2 \pm \sqrt{-4}}{2} = \frac{2 \pm 2i}{2} \]. Simplifying gives \(x = 1 \pm i\). So, the roots are \(x = 1 + i\) and \(x = 1 - i\).

Step 5: List All Zeros of the Polynomial

Combine the zeros from the factored term \(x\) and the quadratic factor. Thus, the zeros of \(P(x)\) are \(0, 1 + i,\) and \(1 - i\).

Step 6: Write the Completely Factored Form of the Polynomial

We express \(P(x)\) using its zeros: \(P(x) = x(x - (1 + i))(x - (1 - i))\). The quadratic term can be rewritten as \((x - 1)^2 + 1^2\) or simply \((x-1)^2 + 1\). Thus, the completely factored form is \(P(x) = x(x^2 - 2x + 2)\).

Key Concepts

Quadratic Formula

The quadratic formula is a key tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is particularly useful when factoring is not straightforward. The formula is written as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this expression:

• \( a \) is the coefficient in front of \( x^2 \)
• \( b \) is the coefficient in front of \( x \)
• \( c \) is the constant term.

By plugging in the values of \( a \), \( b \), and \( c \), you can find the roots of the quadratic equation. In our example, the quadratic part of the polynomial was \( x^2 - 2x + 2 \). Plugging in \( a = 1 \), \( b = -2 \), and \( c = 2 \) allows us to find the zeros. The discriminant \( b^2 - 4ac \) indicates the nature of the roots; if it's negative, the roots are complex. Understanding this formula helps unlock solutions to many polynomial equations.

Complex Numbers

Complex numbers come into play when the discriminant \( b^2 - 4ac \) is negative. A complex number consists of a real part and an imaginary part, typically written in the form \( a + bi \), where \( i \) is the imaginary unit. Imaginary units take the form \( i \), where \( i^2 = -1 \).

• The real part \( a \) and the imaginary part \( bi \) help represent numbers that lie outside the traditional real number line.
• These numbers are vital in factoring polynomials that have no real-number roots.

In the problem at hand, once the quadratic formula revealed a negative discriminant, we knew the roots were complex. The result was the roots \( 1 + i \) and \( 1 - i \), each having a real part (1) and imaginary parts (\( \pm i \)). Mastering complex numbers allows you to solve equations that may seem unsolvable using only real numbers.

Zeros of a Polynomial

The zeros of a polynomial are the solutions to the equation obtained by setting the polynomial to zero. They are also called the roots. For any polynomial \( P(x) \), finding its zeros involves identifying all the values of \( x \) that make \( P(x) = 0 \). These values are crucial as they tell us where the graph of the polynomial crosses or touches the x-axis.

• To find the zeros of a polynomial, you might factor the polynomial, use the quadratic formula for quadratics, or apply other algebraic techniques.
• For the polynomial \( P(x) = x^3 - 2x^2 + 2x \), after factoring out an \( x \) from all terms, we determined zeros as \( x = 0 \), \( x = 1 + i \), and \( x = 1 - i \).

The number of zeros generally equals the degree of the polynomial, though some zeros may be repeated, and others may be complex. Discovering the zeros allows full factoring of the polynomial, giving valuable insights into its behavior and graph.