Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ x^{2}+3 x+4 $$

Step-by-Step Solution

Answer

The polynomial has no real factors over the reals, complex factorization is possible.

Step 1: Identify Coefficients

The polynomial is given as \(x^2 + 3x + 4\). The coefficients are: \(a = 1\), \(b = 3\), and \(c = 4\).

Step 2: Compute the Discriminant

Calculate the discriminant using the formula \(D = b^2 - 4ac\). Substitute the coefficients: \(D = 3^2 - 4 \times 1 \times 4 = 9 - 16 = -7\).

Step 3: Analyze the Discriminant

Since the discriminant \(D = -7\) is negative, the polynomial has no real roots, meaning it's not factorable over the real numbers.

Step 4: Consider Complex Factoring

Since the discriminant is negative, we can find complex roots which can be used to factor the polynomial. Use the quadratic formula to find roots as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 5: Solve for Complex Roots

Using the quadratic formula, substitute in the values: \(x = \frac{-3 \pm \sqrt{-7}}{2}\). The roots are \(x = \frac{-3 + i\sqrt{7}}{2}\) and \(x = \frac{-3 - i\sqrt{7}}{2}\).

Step 6: Write the Complex Factorization

Using the complex roots, the factorization is: \((x - (\frac{-3 + i\sqrt{7}}{2}))(x - (\frac{-3 - i\sqrt{7}}{2}))\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real number system. They include a real part and an imaginary part. The imaginary unit, denoted as \( i \), is defined by the property \( i^2 = -1 \). This means that any number that involves \( \sqrt{-1} \) can be expressed using \( i \). Here is how complex numbers work:

• The standard form of a complex number is \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
• In the context of polynomials, when a quadratic equation has a negative discriminant, the roots are complex.

To solve a polynomial like \( x^2 + 3x + 4 \), where the discriminant is negative, we look for complex roots. Using \( i \) allows us to express these roots as \( x = \frac{-3 \pm i\sqrt{7}}{2} \), showing both the real and imaginary parts.

Discriminant Analysis

Discriminant analysis is a key step in understanding the nature of the solutions to a quadratic equation. The discriminant is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It tells us about the nature of the roots:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root (also known as a repeated or double root).
• If \( D < 0 \), the equation has two complex roots.

For our polynomial \( x^2 + 3x + 4 \), the calculated discriminant \( D = -7 \) is less than zero, indicating complex, rather than real, roots.
This is why factoring over the real numbers is not possible, and we instead use complex numbers to express the roots.

Quadratic Formula

The quadratic formula is used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here is a step-by-step guide on how to apply it:

• Identify the coefficients \( a \), \( b \), and \( c \) in your quadratic equation.
• Substitute these values into the formula.
• Calculate the discriminant \( b^2 - 4ac \) to understand the nature of the roots.
• Solve for \( x \) by performing the arithmetic operations.

When applying the quadratic formula to \( x^2 + 3x + 4 \), and given \( D = -7 \), the roots are expressed in terms of complex numbers. Thus, we compute:
\[x = \frac{-3 \pm i \sqrt{7}}{2}\]
Using this formula, even non-factorable polynomials (over real numbers) can be expressed using roots with imaginary components.