Find The Product Of $3(x+4)(x-5)$.A. $3x^2 - 11x - 20$ B. \$3x^2 - 20x$[/tex\] C. $3x^2 - 3x - 60$ D. $3x^2 - X - 20$

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Introduction

In algebra, multiplying trinomials can be a challenging task, especially when dealing with complex expressions. However, with a clear understanding of the distributive property and the order of operations, we can simplify the process and find the product of a trinomial. In this article, we will explore how to find the product of the expression $3(x+4)(x-5)$ and evaluate the answer choices.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. In the given expression, we have a constant term (3) multiplied by a binomial (x+4) and another binomial (x-5). To find the product, we will apply the distributive property to each term.

Step 1: Multiply the Constant Term with the First Binomial

We will start by multiplying the constant term (3) with the first binomial (x+4). Using the distributive property, we get:

3(x+4)=3x+123(x+4) = 3x + 12

Step 2: Multiply the Result with the Second Binomial

Next, we will multiply the result from Step 1 (3x + 12) with the second binomial (x-5). Again, using the distributive property, we get:

(3x+12)(x−5)=3x(x−5)+12(x−5)(3x + 12)(x-5) = 3x(x-5) + 12(x-5)

Step 3: Apply the Distributive Property Again

Now, we will apply the distributive property to each term inside the parentheses:

3x(x−5)=3x2−15x3x(x-5) = 3x^2 - 15x

12(x−5)=12x−6012(x-5) = 12x - 60

Step 4: Combine Like Terms

Finally, we will combine like terms to simplify the expression:

(3x2−15x)+(12x−60)=3x2−3x−60(3x^2 - 15x) + (12x - 60) = 3x^2 - 3x - 60

Conclusion

After applying the distributive property and combining like terms, we have found the product of the expression $3(x+4)(x-5)$ to be $3x^2 - 3x - 60$. This result matches option C.

Answer Choice Evaluation

Let's evaluate the answer choices to confirm our result:

  • Option A: $3x^2 - 11x - 20$
  • Option B: $3x^2 - 20x$
  • Option C: $3x^2 - 3x - 60$ (Our result)
  • Option D: $3x^2 - x - 20$

Based on our calculation, option C is the correct answer.

Tips and Tricks

When multiplying trinomials, remember to:

  • Apply the distributive property to each term inside the parentheses
  • Combine like terms to simplify the expression
  • Check your answer by evaluating the result with the given options

Q: What is the distributive property, and how is it used in multiplying trinomials?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. When multiplying trinomials, we apply the distributive property to each term inside the parentheses, multiplying each term with the term outside.

Q: How do I apply the distributive property when multiplying trinomials?

A: To apply the distributive property, follow these steps:

  1. Multiply the constant term with the first binomial.
  2. Multiply the result with the second binomial.
  3. Apply the distributive property again to each term inside the parentheses.
  4. Combine like terms to simplify the expression.

Q: What are like terms, and how do I combine them?

A: Like terms are terms that have the same variable and exponent. To combine like terms, add or subtract the coefficients of the like terms.

Q: Can I use the FOIL method to multiply trinomials?

A: Yes, you can use the FOIL method to multiply trinomials. FOIL stands for First, Outer, Inner, Last, and it helps you remember to multiply the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: What are some common mistakes to avoid when multiplying trinomials?

A: Some common mistakes to avoid when multiplying trinomials include:

  • Forgetting to apply the distributive property
  • Not combining like terms
  • Multiplying the wrong terms
  • Not checking the answer with the given options

Q: How do I check my answer when multiplying trinomials?

A: To check your answer, evaluate the result with the given options. Make sure to simplify the expression and combine like terms to ensure that your answer is correct.

Q: Can I use a calculator to multiply trinomials?

A: Yes, you can use a calculator to multiply trinomials. However, it's always a good idea to check your answer with the given options to ensure that it's correct.

Q: What are some real-world applications of multiplying trinomials?

A: Multiplying trinomials has many real-world applications, including:

  • Algebraic geometry
  • Computer science
  • Engineering
  • Physics

Q: Can I use the distributive property to multiply polynomials?

A: Yes, you can use the distributive property to multiply polynomials. However, it's often more efficient to use the FOIL method or other algebraic techniques to simplify the expression.

Conclusion

Multiplying trinomials can be a challenging task, but with the right techniques and practice, you can become proficient in solving these types of problems. Remember to apply the distributive property, combine like terms, and check your answer with the given options to ensure that your result is correct.