Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x^{2}(2 x+3)^{5}(x-4)^{2} $$

Step-by-Step Solution

Answer

Zeros are at \( x = 0 \) (multiplicity 2), \( x = -\frac{3}{2} \) (multiplicity 5), \( x = 4 \) (multiplicity 2).

Step 1: Identify the Factors

The function given is \( f(x)=x^{2}(2x+3)^{5}(x-4)^{2} \). The zeros of the function correspond to the values of \( x \) that make any factor of the product zero. The factors are \( x^{2} \), \( (2x+3)^{5} \), and \( (x-4)^{2} \). We will find the zeros by setting each factor to zero independently.

Step 2: Find the Zero of \( x^2 \)

The factor \( x^2 = 0 \) when \( x = 0 \). This gives us a zero at \( x = 0 \) with multiplicity 2, since the exponent of \( x \) in this factor is 2.

Step 3: Find the Zero of \( (2x+3)^5 \)

Set the factor \( (2x+3)^5 = 0 \) to find the zero. Solve for \( x \):\[2x + 3 = 0 \2x = -3 \x = -\frac{3}{2}\]This gives us a zero at \( x = -\frac{3}{2} \) with multiplicity 5, since the exponent of \( (2x+3) \) is 5.

Step 4: Find the Zero of \( (x-4)^2 \)

Set the factor \( (x-4)^2 = 0 \) to find the zero. Solve for \( x \):\[x - 4 = 0 \x = 4\]This gives us a zero at \( x = 4 \) with multiplicity 2, since the exponent of \( (x-4) \) is 2.

Step 5: Summarize Zeros and Their Multiplicities

The zeros of the function are:- \( x = 0 \) with multiplicity 2- \( x = -\frac{3}{2} \) with multiplicity 5- \( x = 4 \) with multiplicity 2

Key Concepts

Multiplicity of Zeros

In polynomial functions, the **multiplicity of a zero** refers to the number of times a particular zero appears as a root of the polynomial. This determines the behavior of the graph at the zero's location. The multiplicity is directly related to the exponent of the factor that provides the zero. For instance, in the polynomial given by \[ f(x) = x^{2}(2x+3)^{5}(x-4)^{2} \]we have different zeros with distinct multiplicities:

• For \( x^2 \), the zero is at \( x = 0 \) with multiplicity 2, because the exponent of \( x \) in the factor is 2.
• The factor \( (2x+3)^5 \) results in a zero at \( x = -\frac{3}{2} \) with multiplicity 5, since the exponent is 5.
• Lastly, \( (x-4)^2 \) gives a zero at \( x = 4 \) with multiplicity 2, due to its exponent being 2.

A zero with an even multiplicity, such as 2 or 4, implies that the graph of the polynomial touches the x-axis at the zero and then turns around. Conversely, an odd multiplicity, like 3 or 5, indicates that the graph crosses the x-axis.

Factoring Polynomials

**Factoring polynomials** involves breaking them down into their simplest parts, or factors, which are polynomials of lower degrees. This process is crucial for finding the zeros of the polynomial function because the factors set to zero give the roots. For the polynomial function \[ f(x) = x^{2}(2x+3)^{5}(x-4)^{2} \]we can observe that it's already factored into three distinct parts:

• \( x^2 \)
• \( (2x+3)^5 \)
• \( (x-4)^2 \)

Each factor, when equated to zero, gives a zero of the polynomial. Factoring informs us not just of where the zeros occur but also of their multiplicities by looking at the exponents. In practice, polynomials are often expressed in a general expanded form, so being able to correctly factor them is an essential skill in effectively solving polynomials for their zeros.

Finding Zeros of Functions

The process of **finding zeros of functions** involves determining the values of \( x \) that make a polynomial equal to zero. These zeros are also known as roots or solutions of the polynomial function. In the case of \[ f(x) = x^{2}(2x+3)^{5}(x-4)^{2} \]we find its zeros by setting each factor to zero:

• For \( x^2 = 0 \), \( x = 0 \) is a zero.
• From \( (2x+3)^5 = 0 \), solving \( 2x + 3 = 0 \) gives \( x = -\frac{3}{2} \).
• For \( (x-4)^2 = 0 \), \( x = 4 \) is the zero.

Each solution gives a point where the polynomial function crosses or touches the x-axis. Understanding where and how a function meets the x-axis gives insights into the function’s overall behavior. The multiplicity of each zero tells how the graph behaves at each zero, further informing its shape. This process is fundamental in polynomial function analysis.