Problem 81 - Algebra 1

Question

Find the product. $$(2 x-3)(5 x-9)$$

Step-by-Step Solution

Answer

The product of the two binomials (2x-3) and (5x-9) is \(10x^2 - 33x + 27\)

Step 1: Multiply the First terms

First, multiply the first term of the first binomial with the first term of the second binomial. This gives: \(2x * 5x = 10x^2\)

Step 2: Multiply the Outer terms

Then, multiply the first term of the first binomial by the second term of the second binomial (outer terms). This gives: \(2x * -9 = -18x\)

Step 3: Multiply the Inner terms

Next, multiply the second term of the first binomial by the first term of the second binomial (inner terms). This gives: \(-3 * 5x = -15x\)

Step 4: Multiply the Last terms

Lastly, multiply the second term of the first binomial by the second term of the second binomial. This gives: \(-3 * -9 = 27\)

Step 5: Combine the results

Combine the results to get final expression: \(10x^2 - 18x - 15x + 27 = 10x^2 - 33x + 27\).

Key Concepts

Binomial Multiplication

Binomial multiplication is the process of multiplying two binomials together. A binomial is simply an algebraic expression that has two terms, like \((a + b)\). When you multiply two binomials, you have to ensure that each term in the first binomial gets multiplied by each term in the second binomial. Hence, the result will typically be a trinomial or sometimes a polynomial with more terms, depending on how the terms combine.
To perform binomial multiplication effectively, it helps to use a structured approach, such as the distributive property or the FOIL method. These methods systematize the process so you don't miss multiplying any terms.
In our original exercise, we dealt with the expression \((2x - 3)(5x - 9)\). Each term in the first binomial \((2x - 3)\) was multiplied by each term in the second \((5x - 9)\) to form a new polynomial expression.

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions and perform multiplication over addition or subtraction. The property is expressed by the equation: \(a(b + c) = ab + ac\). This means you distribute the multiplication of \(a\) to each term within the parentheses.
In binomial multiplication, you apply this property multiple times to ensure each term is properly multiplied. For example, in the exercise \((2x-3)(5x-9)\), we distribute each term in the first binomial over the terms in the second binomial:

• Multiply \(2x\) with every term in \(5x - 9\).
• Multiply \(-3\) with every term in \(5x - 9\).

As shown in the solution, applying this correctly gives us each individual product (like \(10x^2, -18x, -15x,\) and \(27\)), which we then combine to form the final expression.

FOIL Method

The FOIL method is a specific technique used in binomial multiplication that stands for First, Outer, Inner, Last. It's an easy mnemonic to remember the order in which you multiply the terms of the binomials.
This method ensures that no terms are left out during the multiplication process:

• First: Multiply the first terms of each binomial. For \((2x-3)(5x-9)\), this means multiplying \(2x\) and \(5x\), resulting in \(10x^2\).
• Outer: Multiply the outer terms of the pair of binomials. Here, that is \(2x\) and \(-9\), giving \(-18x\).
• Inner: Multiply the inner terms. In this case, \(-3\) and \(5x\), resulting in \(-15x\).
• Last: Multiply the last terms of each binomial. This is \(-3\) and \(-9\), which gives \(27\).

By following these steps, you ensure a complete and effective multiplication of the binomials, enabling you to combine like terms in the final expression, such as \(10x^2 - 33x + 27\) in our example.