Problem 86 - College Algebra

Question

Perform the indicated operation or operations. $$ (3 x+5)(2 x-9)-(7 x-2)(x-1) $$

Step-by-Step Solution

Answer

The simplified expression for \((3 x+5)(2 x-9)-(7 x-2)(x-1)\) is \(-x^2 - 8x - 47\).

Step 1: Apply FOIL Method to First Pair of Binomials

Using distributive property, expand the first pair of binomials, \((3x+5)(2x-9)\). Multiply each term in the first binomial by each term in the second: \(3x*2x = 6x^2\), \(3x*-9 = -27x\), \(5*2x = 10x\), \(5*-9 = -45\). Combine like terms to obtain \(6x^2-17x-45\).

Step 2: Apply FOIL Method to Second Pair of Binomials

Similarly, expand the second pair of binomials, \((7x-2)(x-1)\): \(7x*x = 7x^2\), \(-2*x = -2x\), \(7x*-1 = -7x\), \(-2*-1 = 2\). Combine like terms to get \(7x^2-9x+2\).

Step 3: Subtract the Two Products

Subtract the result of step 2 from step 1, \(6x^2-17x-45 - (7x^2-9x+2)\). Be careful with the minus sign affecting all terms of the second binomial product. The result will be: \(-x^2 - 8x - 47\).

Key Concepts

Binomials Multiplication

When students first learn to multiply binomials, a technique known as the FOIL method is often introduced. It stands for First, Outer, Inner, Last, which refers to the position of each term in a binomial being multiplied. Simply put, a binomial is an algebraic expression containing two terms, such as \(3x+5\).
To multiply binomials, like in the exercise \(3x+5)(2x-9)\), each term in the first binomial is multiplied by each term in the second binomial. Here’s how it is applied:

• First: Multiply the first terms of each binomial—\(3x * 2x = 6x^2\).
• Outer: Multiply the outer terms—\(3x * -9 = -27x\).
• Inner: Multiply the inner terms—\(5 * 2x = 10x\).
• Last: Multiply the last terms of each binomial—\(5 * -9 = -45\).

After applying these steps, you combine the products to create a single algebraic expression. It’s crucial to keep the order consistent and to apply the correct signs (+ or -) for each term. In the example above, combining the FOIL products results in the expression \(6x^2 - 27x + 10x - 45\).

Distributive Property

A core principle in algebra that is often used in conjunction with the FOIL method is the distributive property. This property allows us to multiply a single term by each term within a parenthesis. In more formal terms, it represents the formula \(a(b + c) = ab + ac\).
In the context of the given exercise, the distributive property is used to expand the product of the binomials. Each term outside the parenthesis is distributed and multiplied by each term inside the parenthesis, which actually forms the basis for the FOIL method. For example, in the second part of the exercise involving \((7x - 2)(x - 1)\), the distributive property looks like this:

• \(7x\) is distributed over \((x - 1)\) resulting in \(7x^2 - 7x\)
• \(-2\) is distributed over \((x - 1)\) resulting in \(-2x + 2\)

After applying the distributive property, it's essential to combine these results to achieve the expanded form. It's crucial for students to understand this property to effectively use the FOIL method and to handle more complex algebraic expressions.

Combining Like Terms

Once you've multiplied the binomials using the FOIL method and have expanded the expression using the distributive property, the next step is to combine like terms. Terms are 'like' if they have the same variable raised to the same power. For example, \(2x\) and \(5x\) are like terms because they both contain the variable \(x\) to the first power.
In our exercise, after using the FOIL method on each binomial pair, we end up with terms that can be combined. For the first product, \(6x^2 - 27x + 10x - 45\), the \(-27x\) and \(10x\) are like terms and can be combined to \(-17x\), resulting in \(6x^2 - 17x - 45\). In the second product, \(7x^2 - 7x - 2x + 2\), the \(-7x\) and \(-2x\) are combined to \(-9x\), resulting in \(7x^2 - 9x + 2\).
After combining like terms in both expanded expressions, they are then subtracted from one another to find the final answer. It’s important to note that when subtracting, we must distribute the negative sign to each term in the second expression, effectively changing their signs before subtracting. This is a common area where mistakes are made, so attention to detail is crucial.