Problem 22 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x-1)(x+2) $$

Step-by-Step Solution

Answer

The product of the binomials \(x-1\) and \(x+2\) is \(x^2 + x - 2\).

Step 1: Identify the elements of the binomials

In our case \(a=x, b=-1, c=x\), and \(d=2\) in the binomials \(x-1\) and \(x+2\) respectively.

Step 2: Apply the product formula

So, applying the formula \(ac+ad+bc+bd\) to the binomials \(x-1\) and \(x+2\), we get \(x*x + x*2 +(-1)*x + (-1)*2\). This simplifies to \(x^2 + 2x - x - 2\)

Step 3: Simplify the expression

Combining like terms, we end up with \(x^2 + x - 2\)

Key Concepts

Binomials

A binomial is a polynomial with exactly two terms. These terms are usually connected by either a plus or minus sign. In the case of our example, the expression \(x - 1\) is a binomial because it contains two terms: \(x\) and \(-1\). Similarly, \(x + 2\) is another binomial with the terms \(x\) and \(+2\). Understanding binomials is crucial because they form the basis of more complex operations involving polynomials.

• Binomials are a type of polynomial.
• They consist of exactly two terms.
• The terms can include variables, constants, or a combination of both.

Recognizing binomials helps us systematically approach problems that involve their multiplication and simplification. Knowing the structure of a binomial can also make applying other mathematical processes, like factoring, more straightforward.

Product Formula

To multiply two binomials, we typically use a product formula. This formula is sometimes remembered through the acronym FOIL, which stands for First, Outer, Inner, Last. But whether using FOIL or a product formula, the core idea is to multiply all combinations of terms between the two binomials. In our example,
\((x - 1)(x + 2)\):

• First: Multiply the first terms in each binomial: \(x \cdot x = x^2\).
• Outer: Multiply the outer terms: \(x \cdot 2 = 2x\).
• Inner: Multiply the inner terms: \(-1 \cdot x = -x\).
• Last: Multiply the last terms in each binomial: \(-1 \cdot 2 = -2\).

This gives us the expression: \(x^2 + 2x - x - 2\). Understanding and applying this formula correctly ensures that all parts of the binomials have been multiplied properly.

Simplifying Expressions

After applying the product formula, we often end up with an expression that can be simplified by combining like terms. In our example, after using the product formula, we arrived at the expression: \(x^2 + 2x - x - 2\). The simplifying stage involves recognizing like terms and combining them:

• Combine \(2x\) and \(-x\) to get \(x\).

Thus, the simplified expression becomes \(x^2 + x - 2\).
Simplifying expressions is a key step in algebra. It helps to make expressions neater and solutions clearer. It involvese:

• Recognizing like terms based on variables and powers.
• Adding or subtracting coefficients of like terms.

Remember, the goal in simplifying is to have the simplest form possible without changing the value of the expression. This aids in both understanding and further manipulation of the equation or expression in broader mathematical contexts.