Problem 42 - Algebra and Trigonometry

Question

Factor the expression completely. $$ z^{2}+6 z-16 $$

Step-by-Step Solution

Answer

The factored form of the expression is \((z + 8)(z - 2)\).

Step 1: Identify the Coefficients

First, identify the coefficients in the quadratic expression. For the expression \( z^2 + 6z - 16 \), the coefficients are \( a = 1 \), \( b = 6 \), and \( c = -16 \).

Step 2: Find Two Numbers that Multiply to ac and Add to b

We need two numbers that multiply to ac (where \( a = 1 \) and \( c = -16 \), so \( ac = -16 \)), and add up to \( b = 6 \). The numbers that satisfy these conditions are 8 and -2 because \( 8 \times -2 = -16 \) and \( 8 + (-2) = 6 \).

Step 3: Rewrite the Middle Term Using the Found Numbers

Rewrite the expression by decomposing the middle term using the two numbers found. The expression becomes \( z^2 + 8z - 2z - 16 \).

Step 4: Factor by Grouping

Group the terms into two pairs: \( (z^2 + 8z) \, \) and \( (-2z - 16) \). Factor out the greatest common factor in each group. From \( z^2 + 8z \), factor out \( z \), giving \( z(z + 8) \). From \( -2z - 16 \), factor out \( -2 \), giving \( -2(z + 8) \).

Step 5: Factor Out the Common Binomial

Both groups now have a common factor of \( (z + 8) \). Factor \( (z + 8) \) out of the expression: \( z(z + 8) - 2(z + 8) = (z + 8)(z - 2) \).

Step 6: Verify the Factored Form

Expand the factored form \( (z + 8)(z - 2) \) to ensure it matches the original expression. Expanding yields \( z^2 - 2z + 8z - 16 \), which simplifies to \( z^2 + 6z - 16 \), confirming the factorization is correct.

Key Concepts

Quadratic Expressions

A quadratic expression is a type of polynomial that is characterized by having a variable raised to the power of two. This is the highest degree in the expression. Typically, a quadratic expression takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants known as coefficients.
Quadratics are essential in algebra because they frequently appear in various mathematical problems, from physics to finance.
Quadratic expressions can have different properties, but the most common aspect is that they have a curved graph called a parabola when plotted on a graph. Understanding how to manipulate and simplify these expressions, such as by factoring, is crucial for solving quadratic equations and analyzing their behavior.

Factor by Grouping

Factor by grouping is a method used to factor certain polynomials, especially quadratic expressions that cannot be immediately factored by common techniques.
The main idea behind this method is to group parts of the expression to find common factors.
To use this method effectively, you perform the following steps:

• Rearrange or rewrite the expression, if necessary, so that similar terms are grouped together.
• Factor out the greatest common factor from each group separately.
• Look for any common binomial factor in the groups, and factor it out.

This technique is powerful for factoring expressions that are not immediately obvious and helps break down the expression into simpler, solvable parts. In the example given, the expression \( z^2 + 6z - 16 \) was rewritten as \( z^2 + 8z - 2z - 16 \), grouped, and successfully factored into two binomials.

Coefficients in Algebra

Coefficients are the numbers placed in front of variables in algebraic expressions. They play a crucial role in understanding and manipulating expressions, especially when it comes to solving equations.
In the quadratic expression given, \( z^2 + 6z - 16 \), the coefficients are \( a = 1 \), \( b = 6 \), and \( c = -16 \).
These coefficients determine the shape and position of the quadratic's graph and influence how the expression can be factored.
Understanding coefficients helps in:

• Performing operations such as addition and subtraction on polynomials.
• Applying different factoring techniques, including factor by grouping.
• Solving quadratic equations using methods like factoring, completing the square, or using the quadratic formula.

By getting comfortable with identifying and using coefficients, you can tackle a wide array of algebraic challenges with more confidence.