Problem 31 - Algebra and Trigonometry

Question

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{4}+3 x^{2}-4\)

Step-by-Step Solution

Answer

Zeros are \(x = 1, x = -1, x = 2i, x = -2i\), each with multiplicity 1.

Step 1: Set Up the Polynomial Equation

The given polynomial is \(P(x) = x^{4} + 3x^{2} - 4\). We need to factor this polynomial completely.

Step 2: Substitution to Simplify Equation

Let's set \(y = x^2\). This substitution transforms the polynomial into a quadratic form: \(y^2 + 3y - 4\).

Step 3: Factor the Quadratic Equation

Now, factor the quadratic \(y^2 + 3y - 4\) into \((y + 4)(y - 1)\).

Step 4: Substitute Back to Original Variable

Replace \(y\) with \(x^2\) to go back to the original variable: \((x^2 + 4)(x^2 - 1)\).

Step 5: Factor Further

Recognize that \(x^2 - 1\) is a difference of squares, so factor it further into \((x - 1)(x + 1)\). Thus, the full factorization is \((x^2 + 4)(x - 1)(x + 1)\).

Step 6: Determine Zeros

Set each factor equal to zero and solve for x:- \(x^2 + 4 = 0\) gives \(x = \pm 2i\)- \(x - 1 = 0\) gives \(x = 1\)- \(x + 1 = 0\) gives \(x = -1\).

Step 7: Determine Multiplicity of Each Zero

Each zero \(x = 1, x = -1, x = 2i, x = -2i\) occurs once, so each has a multiplicity of 1.

Key Concepts

Polynomial Zeros

In the realm of polynomials, finding zeros means identifying the values of \(x\) that make the polynomial equal to zero. These zeros are essential as they reveal critical points where the graph of the polynomial intersects the x-axis. For polynomials like \(P(x) = x^{4} + 3x^{2} - 4\), the task involves factoring the polynomial and then solving for \(x\) in each factor.
Here, once the polynomial is completely factored into \((x^2 + 4)(x - 1)(x + 1)\), determining the zeros becomes straightforward:

• For \(x^2 + 4 = 0\), solving gives \(x = \pm 2i\).
• For \(x - 1 = 0\), solving gives \(x = 1\).
• For \(x + 1 = 0\), solving gives \(x = -1\).

Finding these zeros helps in understanding the behavior of the polynomial function and its graphical representation.

Quadratic Equations

Quadratic equations often appear in the form \(ax^2 + bx + c = 0\) and are a vital part of polynomial math. When factoring a more complex polynomial, substituting to form a quadratic equation can simplify the process significantly. In this exercise, the substitution of \(y = x^2\) transforms the fourth-degree polynomial into the quadratic \(y^2 + 3y - 4\).
This transformation simplifies the task: it allows us to factor using quadratic techniques, such as finding two numbers that multiply to \(-4\) and add to \(3\). Thus, \(y^2 + 3y - 4\) is factored into \((y + 4)(y - 1)\), which is a familiar form making it easier to solve. Quadratics provide a bridge to simpler factoring, ultimately leading us back to the original polynomial context.

Complex Numbers

Complex numbers, involving imaginary units (\(i = \sqrt{-1}\)), are crucial when solving polynomials that lead to non-real solutions. This arises in our factor \((x^2 + 4)\) leading to the equation \(x^2 = -4\). Solving it involves taking the square root of a negative number, which directly brings us to complex numbers.
For \(x^2 + 4 = 0\), the calculation proceeds as:

• Convert \(x^2 = -4\) to \(x = \sqrt{-4} = \pm 2i\).

This result indicates two complex roots: \(2i\) and \(-2i\). Complex numbers extend the real number system and are indispensable in finding all polynomial zeros, especially for degree three or higher.

Zero Multiplicity

The concept of zero multiplicity refers to how many times a zero appears in the factorization of a polynomial. It helps us understand the shape and behavior of the polynomial graph. In practice, multiplicity is determined by the power to which a particular factor is raised. If a zero occurs once, its multiplicity is one.
For \(P(x) = x^{4} + 3x^{2} - 4\), after factoring as \((x^2 + 4)(x - 1)(x + 1)\), the zeros found were \(x = 1, -1, 2i, -2i\). Each factor corresponds to a linear term in the polynomial, indicating a multiplicity of 1 for all zeros:

• \(x = 1\) from \(x - 1\)
• \(x = -1\) from \(x + 1\)
• \(x = 2i, -2i\) from \(x^2 + 4\)

Understanding zero multiplicity aids in predicting the nature of crossings and touches on the x-axis, reflecting the polynomial's graphical movements.