Problem 17 - Algebra and Trigonometry

Question

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$ P(x)=x^{3}+4 x $$

Step-by-Step Solution

Answer

The zeros are 0, 2i, and -2i, each with multiplicity 1.

Step 1: Identify Common Factors

First, identify any common factors that each term of the polynomial shares. For the polynomial \( P(x) = x^3 + 4x \), notice that both terms include a factor of \( x \). We can factor out \( x \) from the expression.

Step 2: Factor Out Common Factor

Factor \( x \) out of the polynomial: \[ P(x) = x^3 + 4x = x(x^2 + 4) \].Now we have factored the polynomial into \( x \times (x^2 + 4) \).

Step 3: Factor the Remaining Polynomial

Next, try to factor \( x^2 + 4 \). However, \( x^2 + 4 \) cannot be factored using real numbers because it does not include any real roots. It can only be factored using complex numbers.

Step 4: Find the Zeros of the Polynomial

To find the zeros, set each factor equal to zero. First, \( x = 0 \) is one zero. For \( x^2 + 4 = 0 \), solve it by setting:\[ x^2 + 4 = 0 \].Solving for \( x \) yields:\[ x^2 = -4 \]\[ x = \pm 2i \].

Step 5: Determine the Multiplicities of Each Zero

The zero \( x = 0 \) has a multiplicity of 1, and the zeros \( x = 2i \) and \( x = -2i \) also each have a multiplicity of 1. No factor is repeated in \( x(x^2 + 4) \), so each zero listed appears once.

Step 6: Conclusion: Complete Factorization and Zeros

The polynomial \( P(x) = x(x^2 + 4) \) is factored completely as over the reals as \( x(x^2 + 4) \) (though \( x^2 + 4 \) cannot be factored further using real numbers). The zeros are \( x = 0, \pm 2i \), each with a multiplicity of 1.

Key Concepts

Complex Numbers

Complex numbers are a fascinating extension of our usual number set, consisting of a real part and an imaginary part. When we say imaginary, we refer to the mathematical unit symbolized by \( i \).
This unit \( i \) has the special property that \( i^2 = -1 \).

• For example, a complex number can be expressed as \( a + bi \), where \( a \) and \( b \) are real numbers.
• The real part is \( a \), and the imaginary part is \( bi \).

Complex numbers become especially useful in solving equations that do not have real solutions. In our case with the equation \( x^2 + 4 = 0 \), we can find solutions or zeros that involve \( \pm 2i \). These zeros are not real, but complex. This is because when solving the equation \( x^2 = -4 \), the negative under the square root requires the use of \( i \).
Therefore, the solutions are \( x = 2i \) and \( x = -2i \). Understanding complex numbers allows mathematicians to solve a broader array of polynomial equations by providing solutions where real numbers fall short.

Polynomial Zeros

Polynomial zeros are values of \( x \) for which the polynomial evaluates to zero. These are critical because they represent the roots or solutions to the equation formed by setting the polynomial equal to zero. For the polynomial \( P(x) = x^3 + 4x \), finding zeros involves factoring the polynomial and identifying when the product of its factors equals zero.

• First, we factor the polynomial to get \( x(x^2 + 4) \).
• This yields potential zeros whenever \( x = 0 \) or when \( x^2 + 4 = 0 \).
• By solving \( x = 0 \), we find one zero as \( x = 0 \).

To find zeros for \( x^2 + 4 = 0 \), we solve for \( x \) using complex numbers.
This yields \( x = 2i \) and \( x = -2i \), thus giving us all the zeros: \( x = 0, \pm 2i \). Polynomials can have zeros in both real and complex domains, adding rich layers to mathematical solutions and investigations.

Multiplicity of Zeros

The multiplicity of a zero in polynomial mathematics refers to the number of times a particular zero appears as a solution. In simpler terms, it tells how many times a polynomial touches or intersects the x-axis at a given point when graphed.

• If a zero has a multiplicity of 1, the curve crosses the x-axis at that point.
• If the multiplicity is even, the graph merely touches the x-axis and turns around.
• If it's odd, the graph crosses through the axis.

For the polynomial \( P(x) = x(x^2 + 4) \), each zero has a multiplicity of 1.

• The zero \( x = 0 \) results from the factor \( x \).
• The zeros \( x = 2i \) and \( x = -2i \) result from the quadratic \( x^2 + 4 \), each with a single occurrence.

Therefore, every root of the polynomial appears just once, signifying each zero's multiplicity is 1.