Problem 23 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 2 z^{2}+20 z+32 $$

Step-by-Step Solution

Answer

Factored form: \(2(z + 2)(z + 8)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at the coefficients of each term in the polynomial \(2z^2 + 20z + 32\). The GCF of 2, 20, and 32 is 2. We will factor 2 out from the trinomial.

Step 2: Factor out the GCF

Divide each term of the trinomial \(2z^2 + 20z + 32\) by the GCF, which is 2. This gives us: \[ 2(z^2 + 10z + 16) \]

Step 3: Factor the Quadratic Expression

Now, focus on the quadratic expression \(z^2 + 10z + 16\). We need two numbers that multiply to 16 and add to 10. These numbers are 2 and 8.

Step 4: Write the Factored Form

Rewrite the quadratic expression \(z^2 + 10z + 16\) using the numbers 2 and 8: \( z^2 + 2z + 8z + 16 \). Factor by grouping: \( z(z + 2) + 8(z + 2) \). Now factor out the common factor \(z + 2\): \( (z + 2)(z + 8) \).

Step 5: Combine with the GCF

Combine the factored quadratic expression with the GCF that was factored out initially. This gives:\[ 2(z + 2)(z + 8) \].

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a critical concept in factoring trinomials. It refers to the largest number that divides all the terms in a polynomial without leaving a remainder. Identifying the GCF is the first step in simplifying expressions and is essential for making further factoring more straightforward.
To find the GCF, you need to look at the coefficients of each term in the trinomial. For example, in the expression \( 2z^2 + 20z + 32 \), the coefficients are 2, 20, and 32. The greatest common number that divides these coefficients is 2.
Once identified, you need to "factor out" the GCF from the expression. This involves dividing each term of the polynomial by 2, which results in a simpler expression: \( 2(z^2 + 10z + 16) \). Factoring out the GCF helps reduce the complexity of the trinomial and lays a solid foundation for further factoring, allowing you to focus on the quadratic portion with just one variable.

Quadratic Expression

A quadratic expression is a type of polynomial where the highest power of the variable is 2. It typically takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratic expressions appear frequently in algebra, and factoring them is a key process for solving many mathematical problems.
In the trinomial we are working with, \( z^2 + 10z + 16 \) is the quadratic expression. The goal is to rewrite this expression into a product of two binomials. This involves finding two numbers that both multiply to the constant term, 16, and add up to the middle coefficient, 10.
The numbers that work for this expression are 2 and 8. When multiplied, they give 16, and when added, they yield 10. Recognizing these numbers allows us to break down the expression into \( z^2 + 2z + 8z + 16 \), paving the way for the next step: factoring by grouping.

Factoring by Grouping

Factoring by grouping is an effective method to factor quadratic expressions when the trinomial cannot easily be factored into a simple pair of binomials. This technique involves grouping terms of the expression to reveal common factors.
In the expression \( z^2 + 2z + 8z + 16 \) (expanded from \( z^2 + 10z + 16 \)), the first two terms \( z^2 + 2z \) can be grouped together, and the last two \( 8z + 16 \) can also be grouped.
From these groupings, we factor out the common factors: \( z(z + 2) + 8(z + 2) \). Noticing that \( (z + 2) \) is a common factor in each group, you can refactor the expression into a product of two binomials: \( (z + 2)(z + 8) \).
Finally, don't forget to incorporate the GCF that was originally factored out in the first step. Combine the result with \( 2 \), the GCF, to obtain the completely factored form: \( 2(z + 2)(z + 8) \). This method simplifies the trinomial and makes it easier to handle or solve.