Problem 46 - Algebra 2

Question

Given a function and one of its zeros, find all of the zeros of the function. \(g(x)=x^{3}+4 x^{2}-27 x-90 ;-3\)

Step-by-Step Solution

Answer

The zeros of the function are \(-3, -6,\) and \(5\).

Step 1: Understand the Problem

We need to find all the zeros of the function \( g(x) = x^3 + 4x^2 - 27x - 90 \). We are given one zero, \( -3 \). Zeros of a function are the values of \( x \) for which \( g(x) = 0 \).

Step 2: Use the Given Zero to Factor the Function

Since \( -3 \) is a zero of \( g(x) \), \( (x + 3) \) is a factor of \( g(x) \). We'll use synthetic division to divide \( g(x) \) by \( (x + 3) \) to find another factor.

Step 3: Perform Synthetic Division

Perform synthetic division:1. Write the coefficients of \( g(x) \): 1, 4, -27, -90.2. Use \( -3 \) for the synthetic divisor.3. Bring down the first coefficient (1).4. Multiply \( -3 \) by this number and add to the next coefficient (4). This gives 1, 1.5. Repeat steps for next coefficients: -23 (for -27) and -21 (for -90).The quotient is \( x^2 + x - 30 \).

Step 4: Factor the Quadratic Quotient

We obtained the quadratic \( x^2 + x - 30 \). We need to factor it to find other zeros.Add factors of -30 that sum to 1 (coefficient of \( x \)): 6 and -5.So, \( x^2 + x - 30 = (x + 6)(x - 5) \).

Step 5: Identify All Zeros of the Function

Using the factors found, the function \( g(x) = (x + 3)(x + 6)(x - 5) \).The zeros are where each factor equal to 0:1. \( x + 3 = 0 \) gives \( x = -3 \)2. \( x + 6 = 0 \) gives \( x = -6 \)3. \( x - 5 = 0 \) gives \( x = 5 \)

Key Concepts

Zero of a Function

Finding the zero of a function is like solving a riddle where you need to discover which numbers make the function equal to zero. Think of a function as a machine: you input a value, and it churns out a result based on its rules, represented by its equation. If the result is zero, you've found a zero of the function.

Zeroes are vital because they indicate where the graph of the function crosses the x-axis. Knowing the zero of a polynomial function like our cubic example, where one zero was given as \(-3\), is the first step in unraveling its complete set of zeroes.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form \((x - c)\). It simplifies work, especially when dealing with higher-degree polynomials. This method requires listing the coefficients of the polynomial and using the zero you already have, which acts as your critical divisor.

Let's break down synthetic division using a simple example. Suppose we have a polynomial and a zero \(c = -3\). Here’s the process:

• Write down the coefficients of the polynomial.
• Use the zero as the divisor.
• Bring down the leading coefficient.
• Multiply the divisor by the leading coefficient you brought down and add the result to the next coefficient.
• Repeat this process for all coefficients.

This results in a new polynomial of one degree lower than the original. Done correctly, synthetic division provides a clearer path to finding the other factors of the polynomial.

Polynomial Factoring

Factoring a polynomial is both an art and a science. After using synthetic division, you are often left with a simpler polynomial to deal with. In our case, we used synthetic division and ended up with a quadratic polynomial \(x^2 + x - 30\). The challenge now is to factor it further into simpler linear factors if possible.

For the quadratic \(x^2 + x - 30\), finding two numbers that multiply to \(-30\) and add up to \(1\) does the trick. In this example, those numbers are \(6\) and \(-5\). Hence, we factor the quadratic as \((x + 6)(x - 5)\).

Successful factoring reveals all zeros of the original polynomial, as these factors can set each equal to zero to solve for all values of \(x\).

Cubic Functions

Cubic functions, expressed generally as \(ax^3 + bx^2 + cx + d\), are polynomial functions of degree three. They have rich properties and exhibit behaviors that are more complex than linear or quadratic functions.

Cubic functions can have up to three distinct real zeros, or roots. For \(g(x) = x^3 + 4x^2 - 27x - 90\), we started by identifying a known zero, \(-3\). From there, through division and factoring, we found the remaining zeros. These zeros indicate where the curve of the cubic graph crosses or touches the x-axis.

Understanding cubic functions opens the door to appreciating more about function behavior, transformations, and the nuances of change rates within mathematical models.