Question

Multiply \((3 x+4)(2 x-5)\) A. \(5 x^2-1\) B. \(6 x^2-20\) C. \(5 x^2-7 x-1\) D. \(6 x^2-7 x-20\)

Short answer

The answer is D. \(6x^2 - 7x - 20\).

Step-by-step solution

- Apply the distributive property

To multiply \((3x+4)(2x-5)\), use the distributive property (also known as the FOIL method for binomials): First, multiply each term in the first binomial by each term in the second binomial.

- Multiply the first terms

Multiply the first terms from each binomial: \(3x \times 2x = 6x^2\).

- Multiply the outer terms

Multiply the outer terms: \(3x \times -5 = -15x\).

- Multiply the inner terms

Multiply the inner terms: \(4 \times 2x = 8x\).

- Multiply the last terms

Multiply the last terms: \(4 \times -5 = -20\).

- Combine like terms

Combine all the terms from the previous steps: \(6x^2 - 15x + 8x - 20\). Combine the like terms \(-15x + 8x\) to get \(-7x\). Therefore, the expression simplifies to \6x^2 - 7x - 20\.

Key concepts

distributive property

To understand how binomial multiplication works, it's crucial to comprehend the distributive property. This property states that to multiply a sum by another number, you need to multiply each addend individually by the number and then sum the results. When you have two binomials, like (3x + 4) and (2x - 5), you apply the distributive property by multiplying each term in the first binomial by each term in the second binomial.

This is popularly known as the FOIL method for binomials:

• First: Multiply the first terms in each binomial. In this case, 3x * 2x = 6x^2.
• Outside: Multiply the outer terms: 3x * -5 = -15x.
• Inside: Multiply the inner terms: 4 * 2x = 8x.
• Last: Multiply the last terms: 4 * -5 = -20.

After distributing, you'll have all the terms that make up the expression before combining like terms.

like terms

Once you've applied the distributive property, the next important step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, 3x and -2x are like terms because they both have the variable x raised to the power of 1. Similarly, 6x^2 and 4x^2 are like terms because the variable x is raised to the power of 2 in both terms.

In our example, after using the FOIL method on (3x + 4)(2x - 5), we get the expression 6x^2 - 15x + 8x - 20. Here, -15x and 8x are like terms. Combining them yields -7x. Always make sure to combine all like terms to simplify the expression fully.

polynomial simplification

The final step in multiplying binomials is polynomial simplification. This is where you combine all the terms to create the simplest form of the polynomial. After multiplying each term and combining like terms, we arrive at the polynomial:

6x^2 - 15x + 8x - 20

We already combined -15x + 8x to get -7x. Now, we can write the simplified expression as:

6x^2 - 7x - 20

Simplifying the polynomial makes it easier to understand and work with, especially when substituting values for xor graphing the polynomial. This final step ensures you have a clean and compact expression for any further calculations.