Expand The Polynomial: { (c-7)(4+2c) - 6c(1-3c) - (9c-2)(3-c)$}$

Introduction

In algebra, expanding a polynomial involves multiplying out the terms to simplify the expression. This process can be time-consuming and error-prone, but with the right techniques, it can be done efficiently. In this article, we will expand the given polynomial expression step by step, using the distributive property and combining like terms.

The Given Polynomial Expression

The given polynomial expression is:

{(c-7)(4+2c) - 6c(1-3c) - (9c-2)(3-c)$}$

Our goal is to expand this expression and simplify it to its final form.

Step 1: Expand the First Term

The first term is $(c-7)(4+2c)$. To expand this term, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. Applying this property, we get:

[$(c-7)(4+2c) = c(4+2c) - 7(4+2c)$

Expanding further, we get:

[$= 4c + 2c^2 - 28 - 14c$

Combining like terms, we get:

[$= 2c^2 - 10c - 28$

Step 2: Expand the Second Term

The second term is $-6c(1-3c)$. To expand this term, we will again use the distributive property:

[$-6c(1-3c) = -6c(1) + 6c(3c)$

Expanding further, we get:

[$= -6c + 18c^2$

Step 3: Expand the Third Term

The third term is $-(9c-2)(3-c)$. To expand this term, we will use the distributive property:

[$-(9c-2)(3-c) = -(9c(3) - 9c(c) - 2(3) + 2(c))$

Expanding further, we get:

[$= -27c + 9c^2 + 6 - 2c$

Combining like terms, we get:

[$= 9c^2 - 29c + 6$

Step 4: Combine the Terms

Now that we have expanded all three terms, we can combine them to get the final expression:

[$(2c^2 - 10c - 28) + (-6c + 18c^2) + (9c^2 - 29c + 6)$

Combining like terms, we get:

[$= 19c^2 - 35c - 22$

Conclusion

In this article, we expanded the given polynomial expression step by step, using the distributive property and combining like terms. We started by expanding the first term, then the second term, and finally the third term. After combining the terms, we got the final expression, which is $19c^2 - 35c - 22$. This expression is the expanded form of the given polynomial.

Key Takeaways

• To expand a polynomial, use the distributive property to multiply out the terms.
• Combine like terms to simplify the expression.
• Use the distributive property to expand each term separately.
• Combine the terms to get the final expression.

Practice Problems

1. Expand the polynomial expression $(2x-3)(x+4) - 5(x-2) - (3x-1)(2-x)$.
2. Expand the polynomial expression $(x+2)(x-3) + 4(x-2) - (2x-1)(x+1)$.
3. Expand the polynomial expression $(3x-2)(x+1) - 2(x-3) - (x-4)(2-x)$.

Solutions

1. $(2x-3)(x+4) - 5(x-2) - (3x-1)(2-x) = 2x^2 + 5x - 12 - 5x + 10 - 6x + 3x - x + 2 = 2x^2 - 2x + 2$
2. $(x+2)(x-3) + 4(x-2) - (2x-1)(x+1) = x^2 - 3x + 2x - 6 + 4x - 8 - 2x^2 - 2x + x + 1 = -x^2 + 3x - 13$
3. $(3x-2)(x+1) - 2(x-3) - (x-4)(2-x) = 3x^2 + 3x - 2x - 2 - 2x + 6 - x^2 + 4x - 2x + 4 = 2x^2 + 3x + 2$

Final Thoughts

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This property allows us to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses.

Q: How do I expand a polynomial expression?

A: To expand a polynomial expression, follow these steps:

1. Identify the terms inside the parentheses.
2. Use the distributive property to multiply each term inside the parentheses by the term outside the parentheses.
3. Combine like terms to simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, if we have the expression $2x + 5x$, we can combine the like terms by adding the coefficients: $2x + 5x = 7x$.

Q: What is the difference between expanding and simplifying a polynomial expression?

A: Expanding a polynomial expression involves multiplying out the terms to simplify the expression. Simplifying a polynomial expression involves combining like terms to reduce the expression to its simplest form.

Q: Can I use a calculator to expand a polynomial expression?

A: Yes, you can use a calculator to expand a polynomial expression. However, it's always a good idea to double-check your work by hand to ensure that the calculator is giving you the correct answer.

Q: How do I know when to use the distributive property?

A: You should use the distributive property whenever you have an expression with parentheses and you need to multiply out the terms.

Q: Can I use the distributive property to expand expressions with more than two terms?

A: Yes, you can use the distributive property to expand expressions with more than two terms. For example, if you have the expression $(a+b+c)(d+e+f)$, you can use the distributive property to expand it as follows: $a(d+e+f) + b(d+e+f) + c(d+e+f)$.

Q: How do I know when to combine like terms?

A: You should combine like terms whenever you have an expression with multiple terms and you can identify like terms.

Q: Can I use a calculator to combine like terms?

A: Yes, you can use a calculator to combine like terms. However, it's always a good idea to double-check your work by hand to ensure that the calculator is giving you the correct answer.

Q: What are some common mistakes to avoid when expanding and simplifying polynomial expressions?

A: Some common mistakes to avoid when expanding and simplifying polynomial expressions include:

• Forgetting to use the distributive property when expanding expressions with parentheses.
• Not combining like terms when simplifying expressions.
• Making errors when multiplying out terms.
• Not checking your work by hand when using a calculator.

Q: How can I practice expanding and simplifying polynomial expressions?

A: You can practice expanding and simplifying polynomial expressions by working through practice problems, such as those found in math textbooks or online resources. You can also try creating your own practice problems to challenge yourself.

Q: What are some real-world applications of expanding and simplifying polynomial expressions?

A: Expanding and simplifying polynomial expressions has many real-world applications, including:

• Algebra: Expanding and simplifying polynomial expressions is a fundamental concept in algebra.
• Calculus: Expanding and simplifying polynomial expressions is used in calculus to solve problems involving rates of change and accumulation.
• Physics: Expanding and simplifying polynomial expressions is used in physics to solve problems involving motion and energy.
• Engineering: Expanding and simplifying polynomial expressions is used in engineering to solve problems involving design and optimization.

Conclusion

Expanding and simplifying polynomial expressions is a fundamental concept in mathematics that has many real-world applications. By understanding the distributive property and how to combine like terms, you can expand and simplify polynomial expressions with ease. Remember to practice regularly to build your skills and confidence.