Problem 47 - 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. $$(6 x+7)(3 x-10)$$

Step-by-Step Solution

Answer

The product is \(18x^2 - 39x - 70\).

Step 1: Identify the Binomial Terms

The problem given is \((6x + 7)(3x - 10)\), which involves two binomials. We'll need to apply the distributive property to expand the product. Identify the two binomials: the first is \((6x + 7)\) and the second is \((3x - 10)\).

Step 2: Apply the FOIL Method

The FOIL method helps us remember to multiply each term in the first binomial by each term in the second binomial. FOIL stands for: First, Outer, Inner, Last. - First: Multiply the first terms: \(6x \cdot 3x = 18x^2\).- Outer: Multiply the outer terms: \(6x \cdot (-10) = -60x\).- Inner: Multiply the inner terms: \(7 \cdot 3x = 21x\).- Last: Multiply the last terms: \(7 \cdot (-10) = -70\).

Step 3: Combine Like Terms

Now, combine the terms from the FOIL method. The equation is:\(18x^2\) (from First) + \(-60x\) (from Outer) + \(21x\) (from Inner) + \(-70\) (from Last).Combine the like terms, which are the linear terms: \(-60x + 21x = -39x\).Thus, the expression becomes \(18x^2 - 39x - 70\).

Key Concepts

Distributive Property

The distributive property is a wonderful tool that lets you multiply a single term across a sum or difference within parentheses. In our binomial multiplication example \((6x + 7)(3x - 10)\), we use this property to expand these expressions into simpler terms.
You might wonder why this method is so essential when multiplying binomials! Well, it allows you to handle each term separately, ensuring no parts are missed during multiplication. This property can be written as: \(a(b+c) = ab + ac\). In our case, you apply this separately to each part of the binomials.

• First, multiply the terms inside one of the binomials by each term in the other binomial.
• Continue this step until every term in the first binomial has been multiplied by every term in the second binomial.

By applying the distributive property, you efficiently expand the product of two binomials into an algebraic expression that is easier to simplify further.

FOIL Method

The FOIL method is an effective shortcut specifically designed for multiplying two binomials quickly. It breaks the multiplication process into four distinct steps that are easier to remember.
FOIL stands for First, Outer, Inner, Last. Let's break down each step:

• First: Multiply the first terms of each binomial. In our example, these terms are \((6x)\) and \((3x)\), resulting in \(18x^2\).
• Outer: Multiply the outer terms, which are \((6x)\) and \((-10)\). This gives \(-60x\).
• Inner: Multiply the inner terms, \((7)\) and \((3x)\), leading to \(21x\).
• Last: Multiply the last terms of each binomial, \((7)\) and \((-10)\), resulting in \(-70\).

By the end of this method, you'll have all the parts you need to combine together to complete your multiplication, making sure that no term is overlooked.

Combining Like Terms

Combining like terms is the next vital step following the use of the FOIL method. After expanding the multiplication, you often find terms with the same variable and exponent. These like terms can be simplified to compress the expression into a neater form.
In our example, after applying the FOIL method, we end up with: \(18x^2 - 60x + 21x - 70\). You'll notice that the terms \(-60x\) and \(21x\) are like terms because they both involve the variable \(x\) raised to the power of 1.

• To combine these, simply add or subtract the coefficients (numbers in front of the variable). Here, you have \(-60x + 21x = -39x\).

This step is crucial for simplifying the expression, which results in the final polynomial: \(18x^2 - 39x - 70\). By combining like terms, you can eliminate unnecessary complexity and present the expression in its simplest form.