Problem 71 - Elementary and Intermediate Algebra

Question

Choose the correct method from Section 6.1 or Section 6.2 to factor each of the following. $$a^{2}-4 a-5$$

Step-by-Step Solution

Answer

The expression \(a^2 - 4a - 5\) factors to \((a - 5)(a + 1)\).

Step 1: Identify the type of quadratic

The given expression is a quadratic polynomial of the form \( a^2 - 4a - 5 \). This indicates that there are coefficients \( a = 1 \), \( b = -4 \), and \( c = -5 \). This expression can be factored by using the method of factoring trinomials, which is covered in Section 6.2.

Step 2: Set up the factorization format

Quadratic expressions of the form \( ax^2 + bx + c \) can generally be expressed as \((x + p)(x + q)\). We need to find values for \( p \) and \( q \) such that \( p + q = b \) (the coefficient of the \( x \) term) and \( p \times q = c \) (the constant term).

Step 3: Find the values of p and q

Now, we look for two numbers whose sum is \( -4 \) and product is \( -5 \). These numbers are \( -5 \) and \( 1 \), because \( -5 + 1 = -4 \) and \( -5 \times 1 = -5 \).

Step 4: Write the factorization

Based on the values we found for \( p \) and \( q \), the factorization of \( a^2 - 4a - 5 \) is \((a - 5)(a + 1)\).

Key Concepts

Quadratic Polynomials

A quadratic polynomial is a type of algebraic expression that involves variables raised to a power of two. It takes the general form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and the variable is usually denoted by \( x \). The leading term is \( ax^2 \), which gives the polynomial its quadratic nature. Quadratic polynomials are fundamental in algebra and appear in many areas of mathematics and applied sciences.

Understanding the components:

• The coefficient \( a \) is known as the leading coefficient and should not be zero, as this would turn the polynomial linear.
• The term \( bx \) is the linear component of the polynomial.
• The constant \( c \) is the standalone number with no associated variable.

Quadratics often represent parabolic graphs when plotted, showing simplicity in symmetry which makes them easier to solve and factor compared to higher degree polynomials.

Factorization involves rewriting the polynomial as a product of linear expressions, continuing into other processes like solving equations or finding roots. This process is crucial for balancing equations, analyzing polynomials, and understanding their graph behavior.

Trinomial Factorization

Trinomial factorization is a technique used to simplify quadratic polynomials by expressing them as a product of two binomial expressions. When dealing with quadratics of the form \( ax^2 + bx + c \), where \( a = 1 \), you focus on finding two numbers that add up to \( b \) and multiply to \( c \).

Consider the example polynomial \( a^2 - 4a - 5 \):

• The sum of two numbers must be equal to \(-4\).
• The product of these same two numbers must be \(-5\).
• Through inspection, \(-5\) and \(1\) satisfy both conditions: \(-5 + 1 = -4\) and \(-5 \times 1 = -5\).

Hence, the factorized form becomes \((a - 5)(a + 1)\).

Trinomial factorization is not only practical for simplifying expressions but also for solving quadratic equations easily. Once in factorized form, it's simple to apply the zero-product property to find solutions by setting each binomial equal to zero.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They do not contain an equality sign, which differentiates them from equations. Understanding algebraic expressions is key to solving and manipulating equations in higher-level mathematics.

Some types of algebraic expressions are:

• Monomials: A single term, like \(3x\).
• Binomials: Two terms, which could be \(x + 5\).
• Trinomials: Three terms, an example being \(x^2 + x + 6\).

These expressions can often be simplified or factored to make them easier to work with. Factoring is an essential skill, especially in algebra, as it transforms complex expressions into simpler components. This process can uncover solutions to equations or provide insights into the properties of the expression.

By breaking down complex expressions into simpler factors or terms, students can work more effectively with polynomials, solve equations, and understand relationships between variables.