Problem 18 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ x^{2}+5 x-6 $$

Step-by-Step Solution

Answer

The expression can be written as \((x - 1)(x + 6)\).

Step 1: Identify Coefficients

Identify the coefficients of the quadratic expression. Here, we have the expression \(x^2 + 5x - 6\), where the coefficient of \(x^2\) is 1, the coefficient of \(x\) is 5, and the constant term is -6.

Step 2: Determine Product and Sum

We need to find two numbers that multiply to give the product of the coefficient of \(x^2\) (which is 1) and the constant term (-6), and also add up to the coefficient of \(x\) (which is 5). This means finding numbers that multiply to -6 and add up to 5.

Step 3: Find Suitable Numbers

List the pairs of numbers that multiply to -6: (1, -6), (-1, 6), (2, -3), and (-2, 3). Out of these, the pair that adds up to 5 is (-1, 6).

Step 4: Write Product of Binomials

Using the numbers found (-1 and 6), we can write the quadratic expression as the product of two binomials: \((x - 1)(x + 6)\).

Step 5: Verify by Expanding

To ensure correctness, expand \((x - 1)(x + 6)\). Using the distributive property (FOIL):\[(x - 1)(x + 6) = x^2 + 6x - x - 6 = x^2 + 5x - 6.\] This confirms our factored form is correct.

Key Concepts

Binomials

In algebra, binomials are expressions made up of two terms connected by either addition or subtraction.
Binomials look like this: \( (a + b) \) or \( (a - b) \). When dealing with quadratics, our goal is often to express a quadratic expression as a product of two binomials.
This process makes it easier to solve equations or to simplify expressions.

• Terms: Parts of the binomial which can be numbers, variables, or a combination of both.
• Operation: Either addition or subtraction connecting the two terms.

For example, the expression \((x - 1)(x + 6)\) consists of two binomials: \(x - 1\) and \(x + 6\).
Factorizing a quadratic into binomials is a key foundation in algebra, especially when solving quadratic equations.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest power of the variable is 2.
They generally take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
These expressions are called "quadratic" because the term "quadratic" means "square," relating to the \(x^2\) term.

• The coefficient of \(x^2\) is what makes it a quadratic. Without this squared term, it would just be linear.
• Quadratic expressions can be factored into binomials, providing an important method for solving them.

In our example, \(x^2 + 5x - 6\), it's shaped perfectly to be factored with the right technique.
The factorization simplifies these expressions, often making them easier to work with.

Distributive Property

The distributive property is a vital algebraic property that allows us to simplify expressions, especially in factorization.
In mathematical terms, it states that for any numbers \(a\), \(b\), and \(c\), we have \(a(b + c) = ab + ac\).
This multiplies through the terms inside the parentheses.

• Helps in expanding products of binomials, ensuring each term in one binomial is multiplied by each term in the other.
• For factorization, allows understanding of the conversion from expanded polynomials back to binomial products.

When verifying factorization, such as in the expression \((x - 1)(x + 6)\), distribute \(x\) and \(-1\) across both \(x\) and \(6\) to simplify back to the original quadratic: \[(x-1)(x+6) = x(x+6) - 1(x+6) = x^2 + 6x - x - 6 = x^2 + 5x - 6\]Understanding and using the distributive property are crucial to mastering algebraic expressions and equations.

FOIL Method

The FOIL method is a specialized application of the distributive property, used solely for multiplying two binomials.
FOIL stands for First, Outer, Inner, and Last, representing the order of multiplication:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of each binomial.

This method simplifies the process of expanding binomial products. For example, if we took the binomials from our quadratic example, \((x - 1)\) and \((x + 6)\), the FOIL method would help us compute:

• First: \(x \cdot x = x^2\)
• Outer: \(x \cdot 6 = 6x\)
• Inner: \(-1 \cdot x = -x\)
• Last: \(-1 \cdot 6 = -6\)

So, combining all parts: \((x - 1)(x + 6) = x^2 + 6x - x - 6 = x^2 + 5x - 6\)The FOIL method is especially helpful for students new to multiplying binomials.