Problem 20

Question

For the following exercises, find the product. $$ (3 d-5)(2 d+9) $$

Step-by-Step Solution

Verified

Answer

$6d^2 + 17d - 45$

1Step 1: Use the Distributive Property

To find the product $(3d-5)(2d+9)$, use the distributive property (also known as the FOIL method for binomials). The first term in each pair of parentheses multiplies with each term in the other pair of parentheses: $(3d)\times(2d)$, $(3d)\times(9)$, $(-5)\times(2d)$, and $(-5)\times(9)$.

2Step 2: Multiply First Terms

First, multiply the first terms: $3d\times2d = 6d^2$. This is the first term of our product.

3Step 3: Multiply Outer Terms

Next, multiply the outer terms: $3d\times9 = 27d$. Add this to our result from Step 2.

4Step 4: Multiply Inner Terms

Now, multiply the inner terms: $-5\times2d = -10d$. Add this to our previous expression.

5Step 5: Multiply Last Terms

Multiply the last terms: $-5\times9 = -45$. Add this to the existing expression.

6Step 6: Combine Like Terms

Having multiplied all pairs, we have $6d^2 + 27d - 10d - 45$. Combine the like terms $27d$ and $-10d$ to simplify: $6d^2 + 17d - 45$.

Key Concepts

FOIL methodMultiplying binomialsCombining like terms

FOIL method

The FOIL method is an acronym to help us remember a specific sequence for multiplying two binomials. FOIL stands for First, Outer, Inner, and Last. These are the four steps you follow:

First: Multiply the first terms in each binomial. For example, with $3d-5$ and $2d+9$, the first terms are $3d$ and $2d$, which gives $6d^2$.
Outer: Multiply the outer terms. Here, those are $3d$ from $3d-5$ and $9$ from $2d+9$, resulting in $27d$.
Inner: Multiply the inside terms. The inside terms are $-5$ and $2d$, giving $-10d$.
Last: Multiply the last terms in each binomial. In this example, this means multiplying $-5$ and $9$ to get $-45$.

The FOIL method is essentially applying the distributive property twice but laid out in a structured manner that simplifies the process. It helps ensure you don't miss any terms.

Multiplying binomials

When you are multiplying binomials, like $3d-5$ and $2d+9$, you are applying the distributive property. This involves multiplying each term in the first binomial by each term in the second binomial.

This process can also be viewed step-by-step:

Multiply the first term of the first binomial by both terms of the second binomial. For $3d$ in $3d-5$, multiply with $2d$ giving $6d^2$, and with $9$ to get $27d$.
Then take the second term of the first binomial, which is $-5$, and multiply it with both terms of the second binomial. Multiply $-5$ with $2d$ to get $-10d$, and $-5$ with $9$ to achieve $-45$.

Following these steps ensures that every term is multiplied correctly, allowing you to form a single expression with all these products combined.

Combining like terms

Once you have multiplied two binomials and gathered all your terms, the next step is to simplify the resulting expression by combining like terms.

Identify Like Terms: Look for terms with the same variable and exponent. In our example, after all multiplications, you have $6d^2 + 27d - 10d - 45$.
Combine Like Terms: Add or subtract coefficients of like terms. Here, $27d$ and $-10d$ are like terms because they both contain the same variable $d$ raised to the first power. Combining them gives $17d$.

The final simplified expression then becomes $6d^2 + 17d - 45$. This step is crucial for presenting your answer in its simplest form.

Problem 20

Other exercises in this chapter

Problem 20

For the following exercises, simplify each expression. $$ 15 \sqrt{5}+7 \sqrt{45} $$

View solution

Problem 20

Multiply the rational expressions and express the product in simplest form. $$ \frac{6 x^{2}-5 x-50}{15 x^{2}-44 x-20} \cdot \frac{20 x^{2}-7 x-6}{2 x^{2}+9 x+1

View solution

Problem 20

Simplify each expression. $$15 \sqrt{5}+7 \sqrt{45}$$

View solution

Problem 20

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. $$ \left(3^{3} \div 3^{4}

View solution