Problem 35 - College Algebra

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=(3 x+2)^{5}\left(x^{2}-10 x+25\right) $$

Step-by-Step Solution

Answer

The zeros are \( x = -\frac{2}{3} \) (multiplicity 5) and \( x = 5 \) (multiplicity 2).

Step 1: Identify the factors

The function given is \( f(x) = (3x + 2)^5 (x^2 - 10x + 25) \). To find the zeros, we need to find when each factor equals zero.

Step 2: Solve for zeros of the first factor

Set the first factor \( (3x + 2) \) to zero: \[ 3x + 2 = 0 \]Solving for \( x \), we subtract 2 from both sides:\[ 3x = -2 \]Then, divide both sides by 3:\[ x = -\frac{2}{3} \]This gives one zero at \( x = -\frac{2}{3} \).

Step 3: Determine the multiplicity of the first zero

The factor \( (3x + 2) \) is raised to the 5th power, which means the zero \( x = -\frac{2}{3} \) has a multiplicity of 5.

Step 4: Solve for zeros of the second factor

Set the second factor \( (x^2 - 10x + 25) \) to zero:\[ x^2 - 10x + 25 = 0 \]Notice this is a perfect square trinomial, which factors as:\[ (x - 5)^2 = 0 \]Solving \( x - 5 = 0 \), we get \( x = 5 \). This gives a zero at \( x = 5 \).

Step 5: Determine the multiplicity of the second zero

The factor \( (x - 5)^2 \) indicates that the zero \( x = 5 \) has a multiplicity of 2.

Key Concepts

Multiplicity of Zeros

When discussing the zeros of a polynomial function, you might hear the term "multiplicity." The multiplicity of a zero refers to the number of times a particular zero appears as a solution of the polynomial equation. In simpler words, it tells you how many times the graph of the polynomial touches or crosses the x-axis at the zero.

• If a zero has a multiplicity of 1, the graph will cross the x-axis at that zero.
• If a zero has a multiplicity of 2, 3, or any other even number, the graph will touch but not cross the x-axis at that zero.
• A multiplicity that is an odd number, other than 1, will cause the graph to cross the x-axis.

Identifying multiplicities is important as it helps us understand the behavior of the graph at specific points. In our exercise, the zero at \( x = -\frac{2}{3} \) has a multiplicity of 5, meaning the graph will touch and turn at this point without crossing the x-axis due to its higher odd multiplicity. Similarly, the zero at \( x = 5 \) has a multiplicity of 2, meaning it will only touch the x-axis.

Factoring Polynomials

Factoring is a vital skill in algebra, especially for finding zeros of polynomial functions. It involves expressing a polynomial as a product of its factors. For example, the polynomial \( f(x) = (3x+2)^5(x^2-10x+25) \) is already partially factored.
To delve deeper, consider the second factor \( (x^2-10x+25) \). This is a perfect square trinomial that can be factored as \( (x-5)^2 \). Perfect square trinomials take the form of \( a^2 \pm 2ab + b^2 \) and can be factored into \( (a \pm b)^2 \). Recognizing this pattern allows us to factor quickly and efficiently.
Factoring simplifies the process of finding zeros because once a polynomial is factored, solving for zeros requires setting each factor to zero. This approach transforms a potentially complex problem into a series of simpler linear problems.

Solving Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). Solving these equations often involves factoring, using the quadratic formula, or completing the square.
In our exercise, one of the steps involved solving \( x^2-10x+25 = 0 \). This particular quadratic equation is a perfect square trinomial, \( (x-5)^2 = 0 \). Recognizing perfect squares simplifies the solution process considerably.Here are common methods to solve quadratic equations:

• Factoring: Works well when the quadratic is factorable into simple binomials, like \( (x-5)^2 \).
• The Quadratic Formula: This method is universal and can solve any quadratic equation: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• Completing the Square: A powerful method, especially when factors are not obvious. It involves rearranging the equation to form a perfect square trinomial.

Each method has its place, and sometimes combining them gives the best results. Understanding each can help solve quadratic equations, no matter how they appear.