Problem 28 - Algebra and Trigonometry

Question

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$ P(x)=x^{5}+7 x^{3} $$

Step-by-Step Solution

Answer

The zeros are \( x = 0 \) (multiplicity 3), \( x = i\sqrt{7} \) (multiplicity 1), \( x = -i\sqrt{7} \) (multiplicity 1).

Step 1: Factor Common Terms

First, look at the polynomial function \( P(x) = x^5 + 7x^3 \). Check for a common factor in each term. Notice that both terms contain the factor \( x^3 \). So, factor out \( x^3 \) from the polynomial:\[ P(x) = x^3(x^2 + 7) \].

Step 2: Identify the Zeros from the Factored Form

Now that we have factored the polynomial as \( P(x) = x^3(x^2 + 7) \), we can identify the zeros. The first zero comes from the factor \( x^3 \), which is \( x = 0 \).

Step 3: Find Additional Zeros from Quadratic

Now look at the factor \( x^2 + 7 \). We need to find values of \( x \) such that \( x^2 + 7 = 0 \). Solving the equation gives us:\[ x^2 = -7 \]\[ x = \pm i\sqrt{7} \].

Step 4: Determine The Multiplicity of Each Zero

Multiplicity refers to the number of times a particular zero appears in the factorized form of the polynomial. For \( x = 0 \), since it's a factor of \( x^3 \), the zero \( x = 0 \) has a multiplicity of 3. For \( x = \pm i\sqrt{7} \), each appears once in the factor, hence each has a multiplicity of 1.

Key Concepts

zeros of a polynomial

Zeros of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as roots or solutions of the polynomial function. Finding these zeros is an essential step when factoring and analyzing polynomials.

When a polynomial is factored completely, the zeros can be easily identified from its factors. For example, if a polynomial is expressed as a product of factors like \( (x-a_1)(x-a_2)...(x-a_n) \), then the zeros are \( a_1, a_2, ... a_n \).

In the given polynomial \( P(x) = x^3(x^2 + 7) \), the zeros are found from each factor:

• From \( x^3 \), we get a zero at \( x = 0 \).
• From \( x^2 + 7 \), solving \( x^2 = -7 \) leads to zeros \( x = \pm i\sqrt{7} \).

Understanding zeros helps in sketching graphs and predicting the behavior of the polynomial function, giving insights into when the function will cross or touch the x-axis.

polynomial multiplicity

Multiplicity in the context of polynomial zeros refers to how many times a particular zero appears as a root of the polynomial. It's a crucial concept in understanding the nature of the roots.

When a zero is repeated, it has a multiplicity greater than one. Multiplicity affects the shape of the polynomial graph, especially around the zero.

• If a zero has an odd multiplicity, the graph crosses the x-axis at that point.
• If a zero has an even multiplicity, the graph only touches the x-axis and turns back.

In the example \( P(x) = x^3(x^2 + 7) \), we observe:

• The zero \( x = 0 \) comes from \( x^3 \), implying it has a multiplicity of 3 because the factor \( x \) is repeated thrice.
• The zeros \( x = \pm i\sqrt{7} \) appear only once, each having a multiplicity of 1.

Recognizing multiplicity is essential for predicting how a polynomial will behave, making it a key feature in higher-level algebra.

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are represented as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part with \( i \) being the imaginary unit, defined by \( i^2 = -1 \).

When dealing with polynomials such as \( x^2 + 7 \), we sometimes come across solutions that are not real numbers. Solving \( x^2 + 7 = 0 \) yielded the solutions \( x = \pm i\sqrt{7} \).

• This is because squaring a real number cannot yield a negative result, leading us to complex solutions.
• The inclusion of complex numbers allows every polynomial equation to have solutions, as stated by the Fundamental Theorem of Algebra.

In mathematics, complex numbers expand our understanding of number systems, allowing for the representation of more intricate mathematical phenomena.