Problem 39 - 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. $$(1+t)(5-2 t)$$

Step-by-Step Solution

Answer

The product is \(-2t^2 + 3t + 5\).

Step 1: Use the Distributive Property

To find the product of two binomials, we'll use the distributive property, also known as the FOIL method (First, Outer, Inner, Last). Start by distributing the first term of the first binomial, which is 1, to each term of the second binomial, giving you \( 1 \times 5 \) and \( 1 \times (-2t) \).

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \( 1 \times 5 = 5 \)

Step 3: Multiply the Outer Terms

Multiply the outer terms: \( 1 \times (- 2t) = -2t \).

Step 4: Multiply the Inner Terms

Now, distribute the second term of the first binomial, \( t \), to each term of the second binomial: \( t \times 5 \) and \( t \times (-2t) \).

Step 5: Multiply the Inner Terms

Multiply the inner terms: \( 5 imes t = 5t \).

Step 6: Multiply the Last Terms

Multiply the last terms: \( t \times (-2t) = -2t^2 \).

Step 7: Combine Like Terms

Gather all the products: \( 5, -2t, 5t, -2t^2 \). Now combine like terms: \( 5t - 2t = 3t \).

Step 8: Write the Final Expression

Write the simplified expression by combining all terms: \(-2t^2 + 3t + 5 \). This is the product of the two binomials.

Key Concepts

Multiplying Binomials

Multiplying binomials is a fundamental concept in algebra. It's important to understand how to combine two binomials to simplify an expression. Binomials are algebraic expressions that contain two terms. For example, in our exercise, we have \((1 + t)\) and \((5 - 2t)\). The goal is to find the product of these two expressions.

One efficient method to multiply binomials is the FOIL method. FOIL is an acronym that stands for First, Outer, Inner, Last. It helps systematically approach the multiplication process.

• First: Multiply the first term of each binomial.
• Outer: Multiply the outermost terms.
• Inner: Multiply the innermost terms.
• Last: Multiply the last term of each binomial.

Using these steps ensures that you properly account for each term's contribution toward the product. This method is particularly useful for students as it lays out a clear path for solving these types of problems.

Distributive Property

The distributive property is a key algebraic property that makes multiplying binomials more manageable. This property states that when you multiply a sum by a number, it's the same as multiplying each addend individually by the number and then summing the products.

In binomial multiplication, each term in the first binomial is distributed over every term in the second binomial. Here's how it works:

• For the binomial \((1 + t)\), you first distribute the '1' by multiplying it with every term in \((5 - 2t)\), so you calculate \(1 \times 5\) and \(1 \times -2t\).
• Then, take the 't' and distribute it similarly: \(t \times 5\) and \(t \times -2t\).

This step-by-step multiplication follows the distributive property. It ensures you don't miss any terms or contributions when multiplying binomials.

Combining Like Terms

After applying the FOIL method and distributive property, you'll have several terms that need simplification. This is where combining like terms becomes crucial. Combining like terms simplifies an expression by adding or subtracting coefficients of terms with the same variable and exponent.

For our example, after distributing, you end up with these terms: \(5, -2t, 5t, -2t^2\). To simplify:

• Recognize like terms: these are the terms that have the same variables raised to the same power. In our expression, \(-2t\) and \(5t\) are like terms because they both contain 't' raised to the power of one.
• Combine the coefficients of like terms: Here, combining \(-2t\) and \(5t\) gives \(3t\).
• Arrange all terms neatly: Place them in decreasing order of their exponents, resulting in \(-2t^2 + 3t + 5\).

Combining like terms is crucial for expressing your final answer in its simplest form, making it clearer and easier to understand.