Problem 24 - Algebra and Trigonometry

Question

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$ P(x)=x^{6}-729 $$

Step-by-Step Solution

Answer

Real zeros are \( x = 3 \) and \( x = -3 \) with multiplicity 1 each.

Step 1: Identify the Difference of Cubes

The polynomial \( P(x) = x^6 - 729 \) can be identified as a difference of squares, where \( x^6 = (x^3)^2 \) and \( 729 = 27^2 \). This can be initially factored as \( (x^3)^2 - 27^2 \), which takes the form of a difference of squares: \( a^2 - b^2 = (a-b)(a+b) \) with \( a = x^3 \) and \( b = 27 \).

Step 2: Apply Difference of Squares Formula

Factor the expression using the difference of squares: \( (x^3 - 27)(x^3 + 27) \).

Step 3: Factor Each Term Further as Difference and Sum of Cubes

Recognize \( x^3 - 27 \) and \( x^3 + 27 \) as a difference and sum of cubes, respectively. These can be factored using the formulas: \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \) and \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \). Here, \( a = x \) and \( b = 3 \).

Step 4: Apply Difference of Cubes Formula

For \( x^3 - 27 \), apply the difference of cubes formula: \( x^3 - 3^3 = (x-3)(x^2 + 3x + 9) \).

Step 5: Apply Sum of Cubes Formula

For \( x^3 + 27 \), apply the sum of cubes formula: \( x^3 + 3^3 = (x+3)(x^2 - 3x + 9) \).

Step 6: Combine Factored Terms

Combine all the factored terms to express \( P(x) \) completely: \( P(x) = (x-3)(x^2 + 3x + 9)(x+3)(x^2 - 3x + 9) \).

Step 7: Find Zeros and Their Multiplicities

Find the zeros by setting each factor equal to zero and solving for \( x \): 1. \( x - 3 = 0 \) gives \( x = 3 \) (multiplicity 1).2. \( x + 3 = 0 \) gives \( x = -3 \) (multiplicity 1).3. \( x^2 + 3x + 9 = 0 \) has no real solutions (use the discriminant \( b^2 - 4ac < 0 \)).4. \( x^2 - 3x + 9 = 0 \) also has no real solutions (use the discriminant \( b^2 - 4ac < 0 \)).

Step 8: State Real Zeros and Multiplicities

The real zeros of the polynomial are \( x = 3 \) and \( x = -3 \), each with multiplicity 1. The other factors do not contribute real zeros because they have complex solutions.

Key Concepts

Difference of Cubes

When dealing with polynomials, the difference of cubes is a useful factoring technique. This method helps in breaking down expressions of the form \(a^3 - b^3\). The polynomial \(x^3 - 27\) is an example of a difference of cubes, where \(a = x\) and \(b = 3\). Using the formula for the difference of cubes, \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\), we can write \(x^3 - 27\) as \((x-3)(x^2 + 3x + 9)\). This factorization reveals the zeros and helps in simplifying polynomial expressions. Remember that this technique only works when both terms are perfect cubes.

Sum of Cubes

The sum of cubes is another important factoring technique that applies to expressions of the form \(a^3 + b^3\). In this exercise, the term \(x^3 + 27\) is a sum of cubes, with \(a = x\) and \(b = 3\). To factor a sum of cubes, we use the formula \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\). For \(x^3 + 27\), the factorization becomes \((x+3)(x^2 - 3x + 9)\). This breakdown shows how expressions can be simplified, and zeros can be identified through factoring.

Zeros of a Polynomial

Zeros of a polynomial are the solutions to the equation formed when setting the polynomial equal to zero. To find the zeros of a factored polynomial, like \(P(x) = (x-3)(x^2 + 3x + 9)(x+3)(x^2 - 3x + 9)\), we set each factor equal to zero:

• \(x - 3 = 0\) gives the zero \(x = 3\).
• \(x + 3 = 0\) gives the zero \(x = -3\).
• \(x^2 + 3x + 9 = 0\) yields complex roots since the discriminant \(b^2 - 4ac\) is negative.
• \(x^2 - 3x + 9 = 0\) also yields complex roots for the same reason.

The real zeros in this case are \(x = 3\) and \(x = -3\). Complex solutions indicate there are no real zeros for these particular terms.

Multiplicity of Zeros

Multiplicity refers to the number of times a zero occurs for a given polynomial. Each distinct zero has an associated multiplicity, which affects the shape of its graph at that point. For \(P(x) = (x-3)(x^2 + 3x + 9)(x+3)(x^2 - 3x + 9)\), the real zeros have a multiplicity of 1. This means that for \(x = 3\) and \(x = -3\), each root appears once. Consequently, the graph of \(P(x)\) will cross the x-axis at these points. Understanding multiplicity is crucial for predicting the behavior of polynomial graphs, as it indicates whether the graph touches or crosses the axis at a zero.