Question

Factor: \(x^{2}-5 x-6\)

Short answer

\[ (x - 6)(x + 1) \]

Step-by-step solution

- Identify the Coefficients

Look at the quadratic equation in the form of \[ ax^2 + bx + c \]In the given equation, \[ x^2 - 5x - 6 \], the coefficients are \( a = 1 \), \( b = -5 \), and \( c = -6 \).

- Find the Product and Sum

To factor the quadratic expression, find two numbers that multiply to \( a \times c \) and add up to \( b \). Here, \( a \times c = 1 \times -6 = -6 \) and \( b = -5 \).Look for two numbers that multiply to -6 and add to -5. These numbers are -6 and 1.

- Rewrite the Middle Term

Rewrite the equation by splitting the middle term using the two numbers found: \[ x^2 - 6x + 1x - 6 \]

- Factor by Grouping

Group the terms into pairs and factor out the greatest common factor (GCF) from each pair:\[ x(x - 6) + 1(x - 6) \]

- Factor the Common Binomial

Factor out the common binomial factor \( (x - 6) \):\[ (x - 6)(x + 1) \]

Key concepts

quadratic equation

A quadratic equation is a polynomial equation of degree 2. This means it has the general form \(ax^2 + bx + c = 0\). Here, \(a, b,\) and \(c\) are constants, with \(a ≠ 0\).
The highest exponent of the variable is 2, which is why it's called 'quadratic'.
Quadratic equations often appear in various topics in math, including physics and engineering problems.

factoring by grouping

Factoring by grouping is a method to solve polynomials and quadratic equations by splitting them into easier parts. Here's how it works:

• First, identify two terms that can be grouped together.
• Next, find the greatest common factor (GCF) of each group.
• Then, factor out the GCF from each group.
• Finally, look for a common binomial factor in the expression.

This method makes it easier to simplify the equation and find its factors.

greatest common factor

The greatest common factor (GCF) is the highest number or expression that divides two or more terms without leaving a remainder. Finding the GCF is essential in factoring by grouping.
For instance, in the equation \(x^2 - 6x + 1x - 6\):

• We group the terms \(x^2 - 6x\) and \(1x - 6\).
• The GCF of \(x^2 - 6x\) is \(x\).
• The GCF of \(1x - 6\) is \(1\).

By factoring out these GCFs, we simplify our polynomials step by step.

polynomials

A polynomial is a mathematical expression consisting of variables (also called indeterminates), coefficients, and exponents. They can take various forms and degrees. For instance:

• A quadratic polynomial: \(ax^2 + bx + c\)
• A cubic polynomial: \(ax^3 + bx^2 + cx + d\)

Polynomials are fundamental in algebra and calculus. They form the basis for various complex equations and functions. Understanding polynomials is crucial for solving many mathematical problems.