Simplify The Expression: ( 3 C − 5 ) ( C + 3 (3c - 5)(c + 3 ( 3 C − 5 ) ( C + 3 ]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving equations and inequalities. It involves combining like terms and removing any unnecessary components from the expression. In this article, we will focus on simplifying the given expression (3c−5)(c+3)(3c - 5)(c + 3) using the distributive property and combining like terms.

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 the parentheses. In the given expression, we have two binomials (3c−5)(3c - 5) and (c+3)(c + 3) that need to be multiplied using the distributive property.

The Distributive Property Formula

(a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd

Using this formula, we can expand the given expression by multiplying each term inside the parentheses with the term outside the parentheses.

Expanding the Expression

To simplify the expression (3c−5)(c+3)(3c - 5)(c + 3), we will use the distributive property to expand it.

Step 1: Multiply the First Term

(3c)(c)=3c2(3c)(c) = 3c^2

Step 2: Multiply the Second Term

(3c)(3)=9c(3c)(3) = 9c

Step 3: Multiply the Third Term

(−5)(c)=−5c(-5)(c) = -5c

Step 4: Multiply the Fourth Term

(−5)(3)=−15(-5)(3) = -15

Combining Like Terms

After expanding the expression, we have the following terms:

3c2+9c−5c−153c^2 + 9c - 5c - 15

Now, we can combine like terms by adding or subtracting the coefficients of the same variables.

Combining Like Terms Formula

a+b+c+d=(a+c)+(b+d)a + b + c + d = (a + c) + (b + d)

Using this formula, we can combine the like terms in the expression.

Simplifying the Expression

After combining like terms, we get:

3c2+4c−153c^2 + 4c - 15

This is the simplified form of the given expression.

Conclusion

In this article, we learned how to simplify the expression (3c−5)(c+3)(3c - 5)(c + 3) using the distributive property and combining like terms. We expanded the expression by multiplying each term inside the parentheses with the term outside the parentheses and then combined like terms to get the simplified form of the expression. This skill is essential in algebra and is used to solve equations and inequalities.

Final Answer

The final answer is: 3c2+4c−15\boxed{3c^2 + 4c - 15}

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Introduction

In our previous article, we learned how to simplify the expression (3c−5)(c+3)(3c - 5)(c + 3) using the distributive property and combining like terms. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional examples to help you practice.

Q&A

Q1: What is the distributive property?

A1: 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 the parentheses.

Q2: How do I apply the distributive property to simplify an expression?

A2: To apply the distributive property, you need to multiply each term inside the parentheses with the term outside the parentheses. For example, in the expression (3c−5)(c+3)(3c - 5)(c + 3), you would multiply 3c3c with cc, 3c3c with 33, −5-5 with cc, and −5-5 with 33.

Q3: What are like terms?

A3: Like terms are terms that have the same variable raised to the same power. For example, in the expression 3c2+4c−5c−153c^2 + 4c - 5c - 15, the terms 4c4c and −5c-5c are like terms because they both have the variable cc raised to the power of 1.

Q4: How do I combine like terms?

A4: To combine like terms, you need to add or subtract the coefficients of the same variables. For example, in the expression 3c2+4c−5c−153c^2 + 4c - 5c - 15, you would combine the like terms 4c4c and −5c-5c by adding their coefficients, which gives you −c-c.

Q5: What is the final answer to the expression (3c−5)(c+3)(3c - 5)(c + 3)?

A5: The final answer to the expression (3c−5)(c+3)(3c - 5)(c + 3) is 3c2+4c−153c^2 + 4c - 15.

Additional Examples

Example 1: Simplify the expression (2x+3)(x−4)(2x + 3)(x - 4)

Solution: (2x)(x)+(2x)(−4)+(3)(x)+(3)(−4)=2x2−8x+3x−12=2x2−5x−12(2x)(x) + (2x)(-4) + (3)(x) + (3)(-4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12

Example 2: Simplify the expression (x−2)(x+5)(x - 2)(x + 5)

Solution: (x)(x)+(x)(5)+(−2)(x)+(−2)(5)=x2+5x−2x−10=x2+3x−10(x)(x) + (x)(5) + (-2)(x) + (-2)(5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10

Example 3: Simplify the expression (3y−2)(y+1)(3y - 2)(y + 1)

Solution: (3y)(y)+(3y)(1)+(−2)(y)+(−2)(1)=3y2+3y−2y−2=3y2+y−2(3y)(y) + (3y)(1) + (-2)(y) + (-2)(1) = 3y^2 + 3y - 2y - 2 = 3y^2 + y - 2

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions and provided additional examples to help you practice. We hope that this article has helped you to understand the concept of simplifying expressions and how to apply it to solve problems.

Final Answer

The final answer is: 3c2+4c−15\boxed{3c^2 + 4c - 15}