Chapter 1: Problem 84
Find the real solutions of each equation by factoring. $$ x^{3}-3 x^{2}-x+3=0 $$
Short Answer
Expert verified
The solutions are \( x = 3 \), \( x = 1 \), and \( x = -1 \).
Step by step solution
01
- Group the terms
Group the terms in pairs to make the equation easier to factor. The given equation is: \[ x^3 - 3x^2 - x + 3 = 0 \]Group terms: \[ (x^3 - 3x^2) + (-x + 3) = 0 \]
02
- Factor by grouping
Factor out the greatest common factor (GCF) from each grouping. From the first group, factor out \( x^2 \) and from the second group factor out \( -1 \): \[ x^2(x - 3) - 1(x - 3) = 0 \]
03
- Factor out the common binomial
Now, factor out the common binomial \( (x - 3) \): \[ (x - 3)(x^2 - 1) = 0 \]
04
- Factor the quadratic expression
The quadratic expression \( x^2 - 1 \) is a difference of squares and can be factored further: \[ (x - 3)(x - 1)(x + 1) = 0 \]
05
- Set each factor to zero and solve
Set each factor equal to zero and solve for \( x \): 1. \( x - 3 = 0 \) 2. \( x - 1 = 0 \) 3. \( x + 1 = 0 \)Solving these gives: 1. \( x = 3 \) 2. \( x = 1 \) 3. \( x = -1 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Real Solutions
Real solutions of a polynomial equation are the values of the variable that satisfy the equation, making it true. These are the points where the polynomial intersects the x-axis on a graph.
To find the real solutions, one can use methods such as factoring, using the quadratic formula, or applying numerical methods. In the context of our exercise, we factor the polynomial to find these solutions. Identifying real solutions is crucial because they reveal the behavior of the polynomial in the real number system and help us understand its graph.
To find the real solutions, one can use methods such as factoring, using the quadratic formula, or applying numerical methods. In the context of our exercise, we factor the polynomial to find these solutions. Identifying real solutions is crucial because they reveal the behavior of the polynomial in the real number system and help us understand its graph.
Factoring by Grouping
Factoring by grouping is a handy technique for polynomials that aren't easily factorable by standard methods.
The process involves grouping terms into pairs and factoring out the greatest common factor (GCF) from each pair.
Let's look at our example: \[ x^3 - 3x^2 - x + 3 = 0 \]
First, we group the terms: \[ (x^3 - 3x^2) + (-x + 3) \]
Then, we factor out the GCF from each group:
Notice we have a common binomial, \(x - 3\). We factor this out, leading us to: \[ (x - 3)(x^2 - 1) \]
The process involves grouping terms into pairs and factoring out the greatest common factor (GCF) from each pair.
Let's look at our example: \[ x^3 - 3x^2 - x + 3 = 0 \]
First, we group the terms: \[ (x^3 - 3x^2) + (-x + 3) \]
Then, we factor out the GCF from each group:
- From the first group, we get: \[ x^2(x - 3) \]
- From the second group, we get: \[ -1(x - 3) \]
Notice we have a common binomial, \(x - 3\). We factor this out, leading us to: \[ (x - 3)(x^2 - 1) \]
Difference of Squares
The difference of squares is a specific type of factoring that applies when you have two squared terms separated by a subtraction sign.
The general formula is: \[ a^2 - b^2 = (a + b)(a - b) \]
In our example, the quadratic expression \( x^2 - 1 \) can be seen as a difference of squares:
Using this step, we can further factor our polynomial equation, leading us to: \[ (x - 3)(x - 1)(x + 1) \]
The general formula is: \[ a^2 - b^2 = (a + b)(a - b) \]
In our example, the quadratic expression \( x^2 - 1 \) can be seen as a difference of squares:
- \( x^2 \) is the square of x
- 1 is the square of 1
Using this step, we can further factor our polynomial equation, leading us to: \[ (x - 3)(x - 1)(x + 1) \]
Quadratic Expression
A quadratic expression is a polynomial of degree 2, typically in the form \( ax^2 + bx + c \). Quadratic expressions are vital in algebra due to their simplicity and the variety of methods available to solve them.
In our factorization process, after factoring by grouping, we encountered \( x^2 - 1 \), which is a quadratic expression. To solve a quadratic expression, you can:
In our factorization process, after factoring by grouping, we encountered \( x^2 - 1 \), which is a quadratic expression. To solve a quadratic expression, you can:
- Factor the expression
- Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Complete the square