Problem 12 - Algebra and Trigonometry

Question

\(5-12\) . Factor out the common factor. $$ (z+2)^{2}-5(z+2) $$

Step-by-Step Solution

Answer

The factored expression is \((z+2)(z-3)\).

Step 1: Identify the common factor

Look at both terms of the expression. Notice that \((z+2)\) is present in both \((z+2)^2\) and \(-5(z+2)\). Hence, \((z+2)\) is the common factor.

Step 2: Rewrite the expression

Express the original expression in terms of the common factor \((z+2)\). The expression becomes \((z+2)((z+2) - 5)\).

Step 3: Simplify the expression

Within the parentheses, perform the subtraction: \((z+2) - 5 = z - 3\). So, the expression now is \((z+2)(z-3)\).

Step 4: Verify your factorization

Distribute the factored terms \((z+2)(z-3)\) to ensure that it equals the original expression. Simplifying \((z+2)(z-3)\) gives \((z+2)^2 - 3(z+2)\), which simplifies to \(z^2 + 4z + 4 - 5z - 10\). This confirms \(z^2 - z - 6\), and distributing back results in the original expression, \((z+2)^2 - 5(z+2)\).

Key Concepts

Common Factor

A common factor is an element that divides each term in an expression without leaving a remainder. Identifying the common factor is the first crucial step when it comes to factoring expressions. In the exercise \((z+2)^2 - 5(z+2)\), the common factor is \((z+2)\) because it appears in both terms:

• \((z+2)^2\)
• \(-5(z+2)\)

By factoring out the common factor, we simplify the expression, making it much easier to work with. This process is essential because it transforms the expression into a format that can be easily solved or further simplified. Identifying and factoring out a common factor helps solve polynomial equations efficiently.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. They involve operations like addition, subtraction, multiplication, and sometimes higher operations like powers.The expression \((z+2)^2 - 5(z+2)\) is a polynomial because it involves a squared term and a linear term.

• The squared term: \((z+2)^2\)
• The linear term: \(-5(z+2)\)

In polynomials, understanding how terms relate is critical. Here, both terms share \((z+2)\), which simplifies the factoring process. Recognizing and manipulating polynomials allows us to rewrite them in simpler forms, especially when expressed in terms of a common factor. This makes solving equations more systematic and manageable.

Simplification

Simplification is the process of rewriting an expression to make it easier to understand or solve. It involves performing operations to reduce complexity.In our exercise, the simplification happens after factoring out the common factor:

• Factor out \((z+2)\) to get \((z+2)((z+2) - 5)\).
• Simplify the equation inside the parentheses: \((z+2) - 5 = z - 3\).
• The expression becomes \((z+2)(z-3)\).

This step reduces the expression to a product of two binomials, which is simpler to interpret and verify. After simplifying, you can check your work by expanding the expression to return to the original to ensure no mistakes were made. Simplification helps uncover solutions and makes complex problems far easier to tackle.