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Chapter 3: Problem 69

a. Factor \(f(x)=9 x^{3}-33 x^{2}+19 x-3,\) given that 3 is a zero. b. Solve. \(9 x^{3}-33 x^{2}+19 x-3=0\)

Short Answer

Expert verified
The solutions are \(x = 3 \) and \(x = \frac{1}{3}\).

Step by step solution

01

Identify the given zero

Given that 3 is a zero of the polynomial, it implies that \(x = 3\) is a root. This means \( (x-3) \) is a factor of \(f(x)=9x^3-33x^2+19x-3\).
02

Perform polynomial division

Divide \( 9x^3-33x^2+19x-3 \) by \( (x-3) \) using synthetic division or long division. You will obtain the quotient as another polynomial.
03

Synthetic division setup

Set up the synthetic division by writing the coefficients of \(9x^3, -33x^2, 19x, -3\) and using \(3\) as the divisor (the zero).
04

Perform synthetic division

Perform the synthetic division: 1. Bring down the leading coefficient (9) unchanged.2. Multiply 9 by the divisor (3) and write the result under the next coefficient (-33).3. Add the results to obtain -6.4. Repeat the process for -6 and 3 to obtain 1, and again for 1 and 3 to obtain 0, indicating no remainder.
05

Write the quotient

The quotient from the synthetic division gives \(9x^2 - 6x + 1\). Therefore, the factorized form of the polynomial \(f(x) \) is \( (x-3)(9x^2 - 6x + 1) \).
06

Solve the equation

To solve \( 9x^3 - 33x^2 + 19x - 3 = 0 \), we set the factors to zero: 1. \(x - 3 = 0\) which gives \(x = 3\).2. Solve \( 9x^2 - 6x + 1 = 0 \) using the quadratic formula \(x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}\).
07

Apply the quadratic formula

For \( 9x^2 - 6x + 1 = 0 \), identify coefficients: \(a = 9, b = -6, c = 1\). Substitute into the quadratic formula: \(x = \frac{6 \pm \sqrt {36-36}}{18} = \frac{6 \pm 0}{18} = \frac{1}{3}\). Thus, the solutions are \(x = \frac{1}{3} \) and \(x = \frac{1}{3} \).
08

Write final solutions

The solutions to \( 9x^3 - 33x^2 + 19x - 3 = 0 \) are \(x = 3 \) and \(x = \frac{1}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is similar to dividing numbers. It involves splitting one polynomial by another. There are two primary methods: long division and synthetic division. Long division follows a similar process to dividing large numbers, where you divide, multiply, subtract, and bring down the next term, continuing until you have processed all terms. Each time, you reduce the degree of the polynomial.

For this exercise, given that 3 is a zero, you can opt for synthetic division, which is generally quicker and simpler for students.
Synthetic Division
Synthetic division is a shortcut method for dividing polynomials when the divisor is a linear factor of the form \(x - c\). It greatly simplifies and speeds up the division process. Here鈥檚 how to perform it:
  • Write down the coefficients of the dividend's polynomial. For example, the coefficients of \(9x^3 - 33x^2 + 19x - 3\) are [9, -33, 19, -3].
  • Place the zero (\

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Most popular questions from this chapter

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