Problem 2 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{2}+11 x+24$$

Step-by-Step Solution

Answer

The polynomial \( x^2 + 11x + 24 \) factors to \( (x + 3)(x + 8) \).

Step 1: Identify the A, B, C coefficients in the trinomial

For the quadratic trinomial given \( x^2 + 11x + 24 \), identify the coefficients: \( A = 1 \), \( B = 11 \), and \( C = 24 \). These coefficients will be used to determine how the trinomial can be factored.

Step 2: Determine potential factor pairs of C

List the factor pairs of the constant term \( C = 24 \). These pairs are: (1, 24), (2, 12), (3, 8), and (4, 6). Look for a pair that can add to equal \( B = 11 \).

Step 3: Check pairs for the correct sum

Examine each factor pair of 24:- \(1 + 24 = 25\) (not 11)- \(2 + 12 = 14\) (not 11)- \(3 + 8 = 11\)Since the pair (3, 8) sums to 11, they are the correct pair to use for factoring.

Step 4: Write the factored form

Now that the correct factor pair (3, 8) has been found, express the quadratic trinomial as a product of binomials using these factors. The factored form is \( (x + 3)(x + 8) \).

Key Concepts

Quadratic Trinomials

A quadratic trinomial is a polynomial with three terms, where the highest degree is two. In simpler terms, it's like a handful of golden rules in algebra that describe how expressions work.
In symbols, it takes the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients. The term "quadratic" signifies that the maximum exponent of the variable "x" is 2. The word "trinomial" just means there are three terms involved, so when you see three connected parts in an expression with a squared term, you're dealing with a trinomial.
When you come across a quadratic trinomial, usually your goal is to find a way to break it down into simpler, easier-to-handle components. This process of breaking down is called factoring, which brings us to the factored form.

Factored Form

The factored form of a quadratic trinomial makes it look simpler and often reveals its roots or solutions. Think of it as turning a complex puzzle into smaller, easier-to-solve pieces by multiplying two binomials.
For our example, \( (x+3)(x+8) \) represents the factored form of \( x^2 + 11x + 24 \). It’s like turning a recipe into a step-by-step guide, where each step is clear and precise.

• The act of factoring helps reveal the original polynomial's zeros or solutions.
• It's efficient for solving quadratic equations or even graphing them.

Factoring transforms equations into a product of linear factors, each binomial multiplying to give the trinomial again when you expand it. This is why finding the right factor pairs is crucial, as these pairs guide the formation of the factored expression.

Integer Factorization

Integer factorization is the process of breaking down numbers into a product of smaller numbers—ideally integers. It's like finding the simplest building blocks to reconstruct a whole.
In our example, the constant term is 24, and finding its integer factors involves listing all the pairs of numbers that multiply to give 24: \((1, 24), (2, 12), (3, 8),\) and \((4, 6)\).
The beauty of integer factorization is that it helps us find ways to express complex numbers or expressions as products of simpler numbers or expressions using integers. It serves as the foundation to simplify expressions, allowing us to break complex mathematical constructs into more manageable parts, which is a vital skill in solving algebraic equations.