Problem 20 - Algebra and Trigonometry

Question

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$ Q(x)=x^{4}-625 $$

Step-by-Step Solution

Answer

Zeros: 5, -5, 5i, -5i. All have multiplicity 1.

Step 1: Recognize the Expression

The given polynomial is \( Q(x) = x^4 - 625 \). Notice that 625 is a perfect square, as \( 625 = 25^2 \). This suggests the expression is a difference of squares.

Step 2: Factor as Difference of Squares

The expression \( x^4 - 625 \) can be rewritten using the formula for the difference of squares: \( a^2 - b^2 = (a-b)(a+b) \). Here, let \( a = x^2 \) and \( b = 25 \), so it becomes \((x^2 - 25)(x^2 + 25) \).

Step 3: Further Factor the Real Difference of Squares

The factor \( x^2 - 25 \) is again a difference of squares, which can be factored further: \( x^2 - 25 = (x-5)(x+5) \).

Step 4: Identify Remaining Terms

The factor \( x^2 + 25 \) is a sum of squares, which does not factor further over the real numbers. However, it can be factored using imaginary numbers: \( x^2 + 25 = (x-5i)(x+5i) \).

Step 5: Write the Complete Factored Form

Combining the factored terms from Steps 3 and 4, the complete factored form of the polynomial \( Q(x) \) is \((x-5)(x+5)(x-5i)(x+5i)\).

Step 6: Find the Zeros and Their Multiplicities

The zeros of the polynomial are identified by setting each factor equal to zero: \( x-5 = 0 \) gives \( x = 5 \) (multiplicity 1), \( x+5 = 0 \) gives \( x = -5 \) (multiplicity 1), \( x-5i = 0 \) gives \( x = 5i \) (multiplicity 1), \( x+5i = 0 \) gives \( x = -5i \) (multiplicity 1).

Key Concepts

Difference of Squares

One of the powerful techniques in polynomial factoring is recognizing and applying the difference of squares. This method applies when a polynomial can be expressed in the form \( a^2 - b^2 \), which is equal to \((a-b)(a+b)\). This pattern is prevalent and, once identified, can simplify the factoring process considerably. In the case of the polynomial \( Q(x) = x^4 - 625 \), the expression is indeed a difference of squares:

• The term \( x^4 \) is the square of \( x^2 \).
• The term \( 625 \) is the square of \( 25 \), since \( 625 = 25^2 \).

By applying the difference of squares formula, the polynomial is rewritten as \((x^2 - 25)(x^2 + 25)\), which simplifies the factoring process and reveals intermediate expressions for further factorization.

Zero Multiplicity

In polynomial expressions, zeros or roots are the values of \( x \) that make the polynomial equal to zero. Each zero has a property known as multiplicity, which indicates how many times a particular root repeats. When examining the polynomial \( Q(x) = (x-5)(x+5)(x-5i)(x+5i) \), each distinct root derived from these factors appears with a multiplicity of 1. This means they are counted once:

• \( x = 5 \) from the factor \( x-5 \).
• \( x = -5 \) from the factor \( x+5 \).
• \( x = 5i \) from the factor \( x-5i \).
• \( x = -5i \) from the factor \( x+5i \).

Recognizing multiplicity is vital because it tells us about the nature of the graph at each zero. A zero with higher multiplicity would cause the graph to "bounce" off the x-axis instead of crossing it.

Complex Zeros

Sometimes, when factoring polynomials, particularly those involving a sum of squares, complex zeros arise. Complex zeros include imaginary numbers and are displayed in the form \( a + bi \), where \( i \) is the imaginary unit defined by \( i^2 = -1 \). For the polynomial \( Q(x) = x^4 - 625 \), when we encounter the term \( x^2 + 25 \), we use complex numbers for further factorization: \( x^2 + 25 = (x-5i)(x+5i) \).

• Here, \( 5i \) and \(-5i\) are complex zeros.
• They illustrate the scenario when the polynomial graph does not intersect the real axis for these roots.

Identifying complex zeros is crucial in fully understanding the nature of a polynomial equation, especially in advanced mathematics, where real-world applications often require considering non-real solutions.