Problem 30 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Check your answer. $$z^{2}-11 z-12$$

Step-by-Step Solution

Answer

The factored form of the expression \(z^2 - 11z - 12\) is \((z + 1)(z - 12)\).

Step 1: Identify the coefficients and constant term

In the given quadratic expression \(z^2 - 11z - 12\), the coefficient of the quadratic term is 1, the coefficient of the linear term is -11, and the constant term is -12.

Step 2: Find two numbers to factorize

We need to find two numbers that multiply to -12 and add to -11. The two numbers that meet these conditions are -12 and 1.

Step 3: Rewrite the expression using the two numbers

Using the two numbers -12 and 1, rewrite the expression: \(z^2 - 11z - 12 = z^2 - 12z + 1z - 12\)

Step 4: Factor by grouping

Now, we will group the terms of the quadratic expression and factor: \(z^2 - 12z + 1z - 12 = (z^2 - 12z) + (1z - 12)\) Factor out the common factors in each group: \((z^2 - 12z) + (1z - 12) = z(z - 12) + 1(z - 12)\)

Step 5: Factor out the common binomial

Since both terms have a common binomial factor of (z - 12), we can factor it out: \(z(z - 12) + 1(z - 12) = (z + 1)(z - 12)\)

Step 6: Check the result

To check our result, we will multiply the factors back: \((z + 1)(z - 12) = z^2 - 12z + 1z - 12 = z^2 - 11z - 12\) The result matches the original expression, so the factored form of \(z^2 - 11z - 12\) is \((z + 1)(z - 12)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are a type of polynomial that have the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The expression consists of three terms:

• The quadratic term \( ax^2 \), which can determine the direction of the parabola's opening.
• The linear term \( bx \), which influences the horizontal shift.
• The constant term \( c \), which affects the vertical position of the graph.

Quadratic expressions can be solved, graphed, or factored, providing various ways to analyze and understand the relationships they represent. A convenient way to express these relationships is by factoring, which allows us to rewrite the quadratic expression in a manner that often reveals its roots or solutions.

Polynomial Factorization

Polynomial factorization is the process of breaking down complex polynomial expressions into a product of simpler polynomials. This process helps in solving polynomial equations and understanding the structure of the polynomials. Here, we encountered the quadratic polynomial \( z^2 - 11z - 12 \).
To factor it, we looked for two numbers that multiply to the constant term, -12, and add to the linear coefficient, -11:

• -12 and 1 multiply to -12
• -12 and 1 add up to -11

These numbers help in rewriting the middle term and facilitate a technique known as 'factor by grouping.' Once terms are grouped, common factors can be extracted, simplifying the expression to its fully factored form. 💡 Pro Tip: This method can be extended to more complex polynomials, providing an effective tool for mathematical problem-solving.

Algebraic Identities

Algebraic identities are equations that hold true for any values of the involved variables. Understanding these identities is beneficial, especially when factoring quadratic expressions.
Some relevant identities include:

• \((a + b)^2 = a^2 + 2ab + b^2\) – Perfect Square Trinomial
• \((a - b)^2 = a^2 - 2ab + b^2\) – Perfect Square Trinomial
• \(a^2 - b^2 = (a - b)(a + b)\) – Difference of Squares

While addressing the expression \( z^2 - 11z - 12 \), the principle of factoring by identifying suitable numbers facilitates use of these identities indirectly. It shows an understanding of how expressions behave without needing to always directly apply an identity, thus making complex factorizations more manageable. Recognizing these patterns simplifies polynomial manipulation and enhances problem-solving efficiency.