Problem 37 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(7 x-2)(2 x+1)$$

Step-by-Step Solution

Answer

The product is \(14x^2 + 3x - 2\).

Step 1: Identify the Binomials

We have two binomials to multiply: \((7x - 2)\) and \((2x + 1)\). The exercise asks us to find their product.

Step 2: Use the FOIL Method

We use the FOIL method to multiply binomials. FOIL stands for First, Outer, Inner, Last, which indicates the pairs of terms we multiply in each binomial.

Step 3: Multiply the First Terms

First, multiply the first terms in each binomial: \( 7x \times 2x = 14x^2 \).

Step 4: Multiply the Outer Terms

Next, multiply the outer terms: \( 7x \times 1 = 7x \).

Step 5: Multiply the Inner Terms

Now, multiply the inner terms: \(-2 \times 2x = -4x \).

Step 6: Multiply the Last Terms

Finally, multiply the last terms: \(-2 \times 1 = -2 \).

Step 7: Combine All Products

Add all the products together: \( 14x^2 + 7x - 4x - 2 \).

Step 8: Simplify the Expression

Combine like terms: \( 14x^2 + 3x - 2 \). This is the simplified product of the binomials.

Key Concepts

FOIL Method

The FOIL Method is an essential tool in algebra for multiplying two binomials. It's a simple acronym that stands for First, Outer, Inner, and Last. This indicates the order in which you multiply the terms of two binomials.
For the given example of \((7x - 2)(2x + 1)\), here's how it works:

• First: Multiply the first terms of each binomial. For \((7x\) and \(2x\), the product is \(7x imes 2x = 14x^2\).
• Outer: Multiply the outer terms: \(7x\) and \(1\). Their product is \(7x\).
• Inner: Multiply the inner terms: \(-2\) and \(2x\), giving \(-4x\).
• Last: Multiply the last terms: \(-2\) and \(1\) which results in \(-2\).

By combining these products, you obtain: \(14x^2 + 7x - 4x - 2\). The FOIL helps ensure that no term is forgotten in the multiplication process.

Polynomial Simplification

Polynomial simplification is the process of combining like terms to make an expression more concise and manageable. After multiplying the binomials, you will often find terms that can be simplified.
In our example \(14x^2 + 7x - 4x - 2\), we need to simplify the terms:

• Identify like terms: These are terms that have the same variable and exponent, in this case, \(7x\) and \(-4x\).
• Combine the coefficients: Add or subtract the coefficients of like terms. So, \(7x - 4x = 3x\).

Now, the simplified expression becomes: \(14x^2 + 3x - 2\). Simplification not only reduces complexity but also aids in further mathematical manipulation and understanding.

Special Products in Algebra

Special products in algebra involve particular patterns when multiplying polynomials. Recognizing these patterns can save time and help avoid errors. Common special products include squares of binomials, difference of squares, and sum or difference of cubes.
Although the given example \((7x - 2)(2x + 1)\) doesn't directly fit into these categories, it's important to be familiar with special products:

• Perfect Square Trinomials: Expressions like \((a + b)^2\) which expand to \(a^2 + 2ab + b^2\).
• Difference of Squares: A pattern like \((a - b)(a + b)\), which simplifies to \(a^2 - b^2\).
• Sum or Difference of Cubes: Patterns like \(a^3 + b^3\) (sum of cubes) or \(a^3 - b^3\) (difference of cubes) have their own set formulas.

Understanding these patterns enables you to quickly solve and verify complex polynomial problems. They are handy tools in your algebra toolkit.