Find The Product And Simplify Your Answer.\[$(z + 2)(4z + 2)\$\]

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Introduction

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Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression: $(z + 2)(4z + 2)$. We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

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The given expression is a product of two binomials: $(z + 2)$ and $(4z + 2)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Applying the Distributive Property

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To simplify the expression, we will apply the distributive property to each term in the first binomial. We will multiply each term in the first binomial by each term in the second binomial.

Step 1: Multiply the First Term in the First Binomial by Each Term in the Second Binomial

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The first term in the first binomial is $z$. We will multiply $z$ by each term in the second binomial: $4z$ and $2$.

$z \cdot 4z = 4z^2$

$z \cdot 2 = 2z$

Step 2: Multiply the Second Term in the First Binomial by Each Term in the Second Binomial

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The second term in the first binomial is $2$. We will multiply $2$ by each term in the second binomial: $4z$ and $2$.

$2 \cdot 4z = 8z$

$2 \cdot 2 = 4$

Combining Like Terms

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Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

$4z^2 + 2z + 8z + 4$

We can combine the like terms $2z$ and $8z$ to get $10z$.

$4z^2 + 10z + 4$

Final Answer

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The simplified expression is $4z^2 + 10z + 4$.

Conclusion

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Simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property and combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we simplified the expression $(z + 2)(4z + 2)$ by applying the distributive property and combining like terms. We hope this article has provided you with a clear understanding of how to simplify algebraic expressions.

Tips and Tricks

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• Always apply the distributive property to each term in the first binomial.
• Combine like terms to simplify the expression.
• Check your work by plugging in values for the variables.

Common Mistakes

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• Failing to apply the distributive property to each term in the first binomial.
• Not combining like terms.
• Not checking your work.

Real-World Applications

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Simplifying algebraic expressions has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.

Practice Problems

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1. Simplify the expression $(x + 3)(2x + 5)$.
2. Simplify the expression $(y - 2)(3y + 1)$.
3. Simplify the expression $(a + 4)(2a - 3)$.

Solutions

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1. $(x + 3)(2x + 5) = 2x^2 + 5x + 6x + 15 = 2x^2 + 11x + 15$
2. $(y - 2)(3y + 1) = 3y^2 + y - 6y - 2 = 3y^2 - 5y - 2$
3. $(a + 4)(2a - 3) = 2a^2 - 3a + 8a - 12 = 2a^2 + 5a - 12$

Conclusion

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Simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property and combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we simplified the expression $(z + 2)(4z + 2)$ by applying the distributive property and combining like terms. We hope this article has provided you with a clear understanding of how to simplify algebraic expressions.

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Q: What is the distributive property?

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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 means that we can multiply a single term by each term in a binomial.

Q: How do I apply the distributive property?

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A: To apply the distributive property, we need to multiply each term in the first binomial by each term in the second binomial. We can do this by following the order of operations (PEMDAS):

1. Multiply the first term in the first binomial by each term in the second binomial.
2. Multiply the second term in the first binomial by each term in the second binomial.
3. Combine like terms.

Q: What are like terms?

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A: Like terms are terms that have the same variable 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?

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A: To combine like terms, we need to 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 are some common mistakes to avoid when simplifying algebraic expressions?

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A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Failing to apply the distributive property to each term in the first binomial.
• Not combining like terms.
• Not checking your work.

Q: How do I check my work when simplifying algebraic expressions?

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A: To check your work when simplifying algebraic expressions, we can plug in values for the variables and see if the expression simplifies to the expected result. For example, if we have the expression $2x + 5x$, we can plug in $x = 1$ and see if the expression simplifies to $7x$.

Q: What are some real-world applications of simplifying algebraic expressions?

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A: Simplifying algebraic expressions has many real-world applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects.
• Engineering: Algebraic expressions are used to design and optimize systems.
• Economics: Algebraic expressions are used to model and analyze economic systems.

Q: How can I practice simplifying algebraic expressions?

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A: There are many ways to practice simplifying algebraic expressions, including:

• Working through practice problems.
• Using online resources and tools.
• Joining a study group or working with a tutor.

Q: What are some tips for simplifying algebraic expressions?

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A: Some tips for simplifying algebraic expressions include:

• Always apply the distributive property to each term in the first binomial.
• Combine like terms to simplify the expression.
• Check your work by plugging in values for the variables.

Q: What are some common algebraic expressions that I should know how to simplify?

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A: Some common algebraic expressions that you should know how to simplify include:

• $(x + a)(x + b)$
• $(x - a)(x - b)$
• $(x + a)(x - b)$

Q: How can I use technology to simplify algebraic expressions?

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A: There are many ways to use technology to simplify algebraic expressions, including:

• Using a graphing calculator to visualize the expression.
• Using a computer algebra system (CAS) to simplify the expression.
• Using online resources and tools to simplify the expression.

Q: What are some advanced topics in algebra that I should know about?

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A: Some advanced topics in algebra that you should know about include:

• Systems of equations.
• Inequalities.
• Functions.
• Graphing.

Q: How can I use algebra to solve real-world problems?

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A: Algebra can be used to solve a wide range of real-world problems, including:

• Physics: Algebra is used to describe the motion of objects.
• Engineering: Algebra is used to design and optimize systems.
• Economics: Algebra is used to model and analyze economic systems.

Q: What are some common mistakes to avoid when using algebra to solve real-world problems?

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A: Some common mistakes to avoid when using algebra to solve real-world problems include:

• Failing to apply the distributive property to each term in the first binomial.
• Not combining like terms.
• Not checking your work.

Q: How can I use algebra to model and analyze real-world systems?

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A: Algebra can be used to model and analyze a wide range of real-world systems, including:

• Physics: Algebra is used to describe the motion of objects.
• Engineering: Algebra is used to design and optimize systems.
• Economics: Algebra is used to model and analyze economic systems.

Q: What are some advanced algebraic techniques that I should know about?

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A: Some advanced algebraic techniques that you should know about include:

• Systems of equations.
• Inequalities.
• Functions.
• Graphing.

Q: How can I use algebra to solve optimization problems?

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A: Algebra can be used to solve a wide range of optimization problems, including:

• Physics: Algebra is used to optimize the motion of objects.
• Engineering: Algebra is used to optimize the design of systems.
• Economics: Algebra is used to optimize economic systems.

Q: What are some common mistakes to avoid when using algebra to solve optimization problems?

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A: Some common mistakes to avoid when using algebra to solve optimization problems include:

• Failing to apply the distributive property to each term in the first binomial.
• Not combining like terms.
• Not checking your work.

Q: How can I use algebra to model and analyze real-world systems?

---

A: Algebra can be used to model and analyze a wide range of real-world systems, including:

• Physics: Algebra is used to describe the motion of objects.
• Engineering: Algebra is used to design and optimize systems.
• Economics: Algebra is used to model and analyze economic systems.

Q: What are some advanced algebraic techniques that I should know about?

---

A: Some advanced algebraic techniques that you should know about include:

• Systems of equations.
• Inequalities.
• Functions.
• Graphing.

Q: How can I use algebra to solve optimization problems?

---

A: Algebra can be used to solve a wide range of optimization problems, including:

• Physics: Algebra is used to optimize the motion of objects.
• Engineering: Algebra is used to optimize the design of systems.
• Economics: Algebra is used to optimize economic systems.

Q: What are some common mistakes to avoid when using algebra to solve optimization problems?

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A: Some common mistakes to avoid when using algebra to solve optimization problems include:

• Failing to apply the distributive property to each term in the first binomial.
• Not combining like terms.
• Not checking your work.

Q: How can I use algebra to model and analyze real-world systems?

---

A: Algebra can be used to model and analyze a wide range of real-world systems, including:

• Physics: Algebra is used to describe the motion of objects.
• Engineering: Algebra is used to design and optimize systems.
• Economics: Algebra is used to model and analyze economic systems.

Q: What are some advanced algebraic techniques that I should know about?

---

A: Some advanced algebraic techniques that you should know about include:

• Systems of equations.
• Inequalities.
• Functions.
• Graphing.

Q: How can I use algebra to solve optimization problems?

---

A: Algebra can be used to solve a wide range of optimization problems, including:

• Physics: Algebra is used to optimize the motion of objects.
• Engineering: Algebra is used to optimize the design of systems.
• Economics: Algebra is used to optimize economic systems.

Q: What are some common mistakes to avoid when using algebra to solve optimization problems?

---

A: Some common mistakes to avoid when using algebra to solve optimization problems include:

• Failing to apply the distributive property to each term in the first binomial.
• Not combining like terms.
• Not checking your work.

Q: How can I use algebra to model and analyze real-world systems?

---

A: Algebra can be used to model and analyze a wide range of real-world systems, including:

• Physics: Algebra is used to describe the motion of objects.
• Engineering: Algebra is used to design and optimize systems.
• Economics: Algebra is used to model and analyze economic systems.

Q: What are some advanced algebraic techniques that I should know about?

---

A: Some advanced algebraic techniques that you should know about include:

• Systems of equations.
• Inequalities.
• Functions.
• Graphing.

**Q: How can I use algebra to