Problem 14 - Algebra 1: Concepts and Skills

Question

Choose the correct factorization. $$ x^{2}+11 x-26 $$ $$ \begin{aligned} &a.\quad(x-13)(x+2)\\\ &b.\quad(x+13)(x-2) \end{aligned} $$

Step-by-Step Solution

Answer

The correct factorization of the quadratic equation \(x^{2}+11 x-26\) is (b) \((x+13)(x-2)\).

Step 1: Identify the values

The quadratic equation is given by \(x^{2}+11 x-26\), which is the standard form \(ax^{2}+bx+c\). From this, we identify that \(a=1\), \(b=11\), and \(c=-26\).

Step 2: Determine possible pairs

We need to find two numbers that multiply to give \(ac=-26\) and add to give \(b=11\). These two numbers would be the constants in the binomial expressions that are the factorization of the given quadratic equation.

Step 3: Check the given options

\nOption a: \((x-13)(x+2)\): If we FOIL this, we get \(x^{2}-13x+2x-26\) which simplifies to \(x^{2}-11x-26\), this is not equal to the given equation. \nOption b: \((x+13)(x-2)\): If we FOIL this, we get \(x^{2}+13x-2x-26\) which simplifies to \(x^{2}+11x-26\), this is equal to the given equation. \nTherefore, option b is the correct factorization of the given quadratic equation.

Key Concepts

Factoring Quadratics

When you come across a quadratic equation, \textbf{factoring} is a method you can use to solve it. Factoring quadratics involves breaking down the equation into a product of binomials. Let's say we have the quadratic equation \(x^2 + 11x - 26\). To factor this, you look for two numbers that multiply to the constant term (\(c\)) and add up to the linear coefficient (\(b\)). Here, we want two numbers that multiply to -26 and add up to 11. After finding the numbers, you would then write them as binomial expressions that multiply to form the original quadratic equation.

For the equation above, the factors of -26 that add up to 11 are 13 and -2. This means we can write the quadratic as the product of two binomials: \(x + 13\) and \(x - 2\). Therefore, the correct factorization is \[\(b.\quad(x+13)(x-2)\)\]. Factoring is a crucial skill in algebra, as it appears in many areas of mathematics and applied problems.

FOIL Method

The \textbf{FOIL method} is a handy technique used to multiply two binomial expressions. FOIL stands for First, Outer, Inner, Last, which are the terms you multiply together when using this method. To see how it works, let's multiply the binomials \(x + 13\) and \(x - 2\) as in our factorization example.

• The \textbf{F}irst terms are \(x\) and \(x\), multiplied to give \(x^2\).
• The \textbf{O}uter terms are \(x\) and \( -2\), giving \( -2x\).
• The \textbf{I}nner terms are \(13\) and \(x\), giving \(13x\).
• The \textbf{L}ast terms are \(13\) and \( -2\), which give \( -26\).

When you add up all these products, you get \(x^2 + 11x - 26\), which is what we want. The FOIL method is a great tool for checking your work when you factor quadratics and is essential for understanding how to expand binomials.

Binomial Expressions

A \textbf{binomial expression} is an algebraic expression containing two terms. For example, \(x + 13\) and \(x - 2\) are both binomials. Factoring quadratic equations often results in two binomial expressions. Understanding how to handle binomials is key when learning to factor quadratics or expand polynomials using the FOIL method.

Each term in a binomial may consist of variables, numbers, or a combination of both. They are separated by a plus (\( + \) ) or minus (\( - \) ) sign. When factoring, you are reverse-engineering the multiplication process of binomials to find the simplest expressions that would multiply together to give you the original quadratic equation. Remember, the power of binomials lies in their simplicity - they are much easier to work with compared to more complex expressions.