Problem 36 - College Algebra

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x\left(4 x^{2}-12 x+9\right)\left(x^{2}+8 x+16\right) $$

Step-by-Step Solution

Answer

Zeros are \( x = 0 \) (multiplicity 1), \( x = \frac{3}{2} \) (multiplicity 2), \( x = -4 \) (multiplicity 2).

Step 1: Identify the Factors

First, look at the expression given: \( f(x) = x(4x^2 - 12x + 9)(x^2 + 8x + 16) \). The function is already factored into three parts: \( x \), \( 4x^2 - 12x + 9 \), and \( x^2 + 8x + 16 \).

Step 2: Find Zeros of Each Factor

To find the zeros, set each factor equal to zero and solve for \( x \).1. For \( x \): \( x = 0 \).2. For \( 4x^2 - 12x + 9 \): Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = -12 \), and \( c = 9 \). \( b^2 - 4ac = (-12)^2 - 4 \cdot 4 \cdot 9 = 144 - 144 = 0 \) There is one real solution since the discriminant is 0: \( x = \frac{-(-12)}{2 \cdot 4} = \frac{12}{8} = \frac{3}{2} \).3. For \( x^2 + 8x + 16 \): Also, use the quadratic formula with \( a = 1 \), \( b = 8 \), \( c = 16 \). \( b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot 16 = 64 - 64 = 0 \) One real solution for \( x \): \( x = \frac{-8}{2} = -4 \).

Step 3: Determine Multiplicities

Multiplicity refers to the number of times a particular zero appears in the factorization:1. The zero \( x = 0 \) comes from the factor \( x \), so it has a multiplicity of 1.2. The zero \( x = \frac{3}{2} \) is the root of the perfect square trinomial \( (2x - 3)^2 = 4x^2 - 12x + 9 \), so its multiplicity is 2.3. The zero \( x = -4 \) is the root of the perfect square trinomial \( (x + 4)^2 = x^2 + 8x + 16 \), so its multiplicity is 2.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions, or "zeros," of a quadratic equation. A quadratic equation is generally written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). If you're scratching your head about how to find a zero, think of solving an equation to find where the graph of the quadratic touches the x-axis. This touchpoint, also called a 'root', can be found using this straightforward formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides two solutions, one for the plus sign and another for the minus. These solutions could be real or complex numbers, depending on the discriminant participation inside the formula. Remember, this single equation carries all the potential outcomes for our quadratic conundrum; that's why it’s so valuable.

Discriminant

When dealing with the quadratic formula, the discriminant is a crucial part. It is the expression under the square root: \( b^2 - 4ac \). The discriminant helps determine the nature of the roots of the quadratic equation without solving the entire formula.

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real solutions.
• If \( b^2 - 4ac = 0 \), it has exactly one real solution (a repeated root).
• If \( b^2 - 4ac < 0 \), there are no real solutions — the roots are complex numbers.

In our original problem, for each quadratic expression, the discriminants were 0. This indicated that each quadratic had a single repeated root, meaning the graph of the quadratic just "kisses" the x-axis without fully crossing it, forming a "bounce." Understanding the discriminant deepens your insight into the behavior of quadratic equations and prepares you for what solutions to expect.

Multiplicity of a Root

Once you find the roots of a polynomial, it's important to determine the "multiplicity" of each root. Multiplicity refers to how many times a particular root appears in the factorization of the polynomial. This can affect the graph's shape at the root location.

• A root with multiplicity 1 means the graph simply crosses the x-axis at that point.
• If the root has an even multiplicity (like 2 or 4), the graph "bounces" off the x-axis at the root.
• An odd multiplicity greater than 1 (like 3, 5, etc.) indicates that the curve flattens at the x-axis and then crosses.

In our exercise, the roots \( x = \frac{3}{2} \) and \( x = -4 \) both have a multiplicity of 2. This means the graph of the polynomial bounces off the x-axis at these points, without crossing it. While the root \( x = 0 \) has a multiplicity of 1, indicating a simple crossing, highlighting this detail ensures you're not just solving equations but also visualizing their graphical implications.