Problem 91 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. It's easy to factor \(x^{2}+x+1\) because of the relatively small numbers for the constant term and the coefficient of \(x\)

Step-by-Step Solution

Answer

The statement 'It's easy to factor \(x^{2}+x+1\) because of the relatively small numbers for the constant term and the coefficient of \(x\)' does not make sense. This is because the quadratic \(x^{2}+x+1\) cannot be factored over the set of real numbers due to a negative discriminant.

Step 1: Understand the Statement

Analyze the statement 'It's easy to factor \(x^{2}+x+1\) because of the relatively small numbers for the constant term and the coefficient of \(x'\). The statement is suggesting that the quadratic equation \(x^{2}+x+1\) can be easily factored based solely on the relatively small size of its coefficients.

Step 2: Apply Quadratic Factoring Rules

To see if the statement is valid, apply the method for factoring quadratic equations. By using the quadratic formula \((-b\pm\sqrt{b^2-4ac})/2a\) where \(a=1\), \(b=1\), and \(c=1\), it should calculate the roots. If the roots are real, the quadratic can be factored

Step 3: Calculate The Discriminant

Calculate the discriminant \(b^{2}-4{{ac}}\) from the quadratic formula. The discriminant is \(1^{2}-4(1)(1) = 1 - 4 = -3\).

Step 4: Evaluate The Discriminant

Since the discriminant is negative, the roots of the equation are not real numbers. Because the equation has no real roots, it cannot be factored over the set of real numbers.

Key Concepts

quadratic formula

The quadratic formula is a mathematical tool used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula provides a straightforward way to find the roots of these equations, particularly when factoring is difficult or impossible. The general formula is:

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

This formula helps us find the values of \( x \) that satisfy the equation. It's important to first identify the coefficients \( a \), \( b \), and \( c \) in the equation.

For example, in the expression \( x^2 + x + 1 \), the coefficients are \( a = 1 \), \( b = 1 \), and \( c = 1 \). Inserting these into the quadratic formula will enable us to determine the nature of the roots. Understanding these coefficients' role is crucial for solving the equation using this method.

discriminant

The discriminant is a part of the quadratic formula that helps you analyze the nature of the roots without solving the equation completely. It is given by the expression \( b^2 - 4ac \).

• If the discriminant is positive, there are two distinct real roots.
• If it equals zero, there is exactly one real root, often called a repeated root.
• If it's negative, there are no real roots, but rather two complex roots.

In our problem, the discriminant is calculated as \( 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3 \).

Since the discriminant is negative, it indicates that our quadratic equation does not have real roots. This shows why factoring over the set of real numbers is not possible in this case, emphasizing the significance of the discriminant in quickly determining the type of roots.

real and complex roots

Understanding real and complex roots is essential when working with quadratic equations. Real roots are the solutions of the equation that can be plotted on a real number line, whereas complex roots involve imaginary numbers and cannot be located on the real number line.

• Real roots occur when the discriminant is zero or positive.
• Complex roots arise when the discriminant is negative.

When a quadratic equation like \( x^2 + x + 1 \) turns out to have a negative discriminant (\(-3\) in this case), it indicates the presence of two complex roots.

These complex roots are conjugates of each other, expressed in the form \( a + bi \) and \( a - bi \), where \( i \) is the imaginary unit. Recognizing whether roots are real or complex allows one to choose the appropriate method for solving the quadratic equation.