{ (x-3)\left(x^2-2x+4\right)$}$What Is The Coefficient Of { X$}$ When The Expression Is Expanded?

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Introduction

In algebra, expanding expressions is a crucial step in solving equations and manipulating mathematical statements. When dealing with polynomials, it's essential to understand how to expand them to simplify and analyze the expression. In this article, we will focus on expanding the given expression {(x-3)\left(x^2-2x+4\right)$}$ and find the coefficient of {x$}$.

Understanding the Expression

The given expression is a product of two binomials: {(x-3)$}$ and {\left(x^2-2x+4\right)$}$. To expand 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

To expand the given expression, we will multiply each term in the first binomial by each term in the second binomial. This will result in a sum of products, which can be simplified to obtain the expanded expression.

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

The first term in the first binomial is {x$}$. We will multiply this term by each term in the second binomial: {x^2$, [-2x$, and [4\$}.

Multiplying {x$}$ by {x^2$

When we multiply [x\$} by {x^2$}$, we get {x^3$}$.

Multiplying {x$}$ by {-2x\$

When we multiply [x\$} by βˆ’2x${-2x\$}, we get βˆ’2x2${-2x^2\$}.

Multiplying {x$}$ by ${4\$}

When we multiply {x$}$ by ${4\$}, we get ${4x\$}.

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

The second term in the first binomial is βˆ’3${-3\$}. We will multiply this term by each term in the second binomial: {x^2$, [-2x$, and [4\$}.

Multiplying βˆ’3${-3\$} by {x^2$

When we multiply [-3$}$ by {x^2$}$, we get βˆ’3x2${-3x^2\$}.

Multiplying βˆ’3${-3\$} by {-2x\$

When we multiply [-3$}$ by βˆ’2x${-2x\$}, we get 6x${6x\$}.

Multiplying βˆ’3${-3\$} by ${4\$}

When we multiply βˆ’3${-3\$} by ${4\$}, we get βˆ’12${-12\$}.

Combining Like Terms

Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms to simplify the expression.

Combining the Terms with {x^3$}$

The only term with {x^3$}$ is {x^3$}$ itself.

Combining the Terms with βˆ’2x2${-2x^2\$}

The terms with βˆ’2x2${-2x^2\$} are βˆ’2x2${-2x^2\$} and βˆ’3x2${-3x^2\$}. Combining these terms, we get βˆ’5x2${-5x^2\$}.

Combining the Terms with ${4x\$}

The terms with ${4x\$} are ${4x\$} and 6x${6x\$}. Combining these terms, we get 10x${10x\$}.

Combining the Constant Terms

The constant terms are 0${0\$} and βˆ’12${-12\$}. Combining these terms, we get βˆ’12${-12\$}.

The Expanded Expression

After combining like terms, the expanded expression is {x^3 - 5x^2 + 10x - 12$}$.

Finding the Coefficient of {x$}$

To find the coefficient of {x$}$, we need to look at the term with {x$}$ in the expanded expression. In this case, the term with {x$}$ is 10x${10x\$}. The coefficient of {x$}$ is the number that multiplies {x$}$, which is 10${10\$}.

Conclusion

In this article, we expanded the given expression {(x-3)\left(x^2-2x+4\right)$}$ and found the coefficient of {x$}$. The expanded expression is {x^3 - 5x^2 + 10x - 12$}$, and the coefficient of {x$}$ is 10${10\$}. This demonstrates the importance of expanding algebraic expressions to simplify and analyze mathematical statements.

Q: What is the distributive property, and how is it used in expanding algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, {a(b+c) = ab + ac$}$. This property is used to expand algebraic expressions by multiplying each term in one binomial by each term in the other binomial.

Q: How do I apply the distributive property to expand an algebraic expression?

A: To apply the distributive property, you need to multiply each term in one binomial by each term in the other binomial. This will result in a sum of products, which can be simplified to obtain the expanded expression.

Q: What is the difference between a binomial and a polynomial?

A: A binomial is an algebraic expression consisting of two terms, such as {x + 3$}$ or {x^2 - 2x + 4$}$. A polynomial, on the other hand, is an algebraic expression consisting of three or more terms, such as {x^3 - 5x^2 + 10x - 12$}$.

Q: How do I combine like terms in an expanded expression?

A: To combine like terms, you need to identify the terms with the same variable and exponent, and then add or subtract their coefficients. For example, in the expression {x^3 - 5x^2 + 10x - 12$}$, the terms with βˆ’5x2${-5x^2\$} and 10x${10x\$} can be combined to get βˆ’5x2+10x${-5x^2 + 10x\$}.

Q: What is the coefficient of a term in an algebraic expression?

A: The coefficient of a term is the number that multiplies the variable. For example, in the expression {x^3 - 5x^2 + 10x - 12$}$, the coefficient of {x^3$}$ is 1$,thecoefficientof\[βˆ’5x2${1\$, the coefficient of \[-5x^2\$} is βˆ’5$,andthecoefficientof\[10x${-5\$, and the coefficient of \[10x\$} is 10${10\$}.

Q: How do I find the coefficient of a specific term in an expanded expression?

A: To find the coefficient of a specific term, you need to identify the term with the variable and exponent you are interested in, and then look at the number that multiplies the variable. For example, in the expression {x^3 - 5x^2 + 10x - 12$}$, the coefficient of {x$}$ is 10${10\$}.

Q: What is the importance of expanding algebraic expressions?

A: Expanding algebraic expressions is an essential step in solving equations and manipulating mathematical statements. By expanding expressions, you can simplify and analyze the expression, which can help you solve problems and make mathematical statements more clear and concise.

Q: Can you provide examples of real-world applications of expanding algebraic expressions?

A: Yes, expanding algebraic expressions has many real-world applications, such as:

  • Physics and Engineering: Expanding algebraic expressions is used to describe the motion of objects, calculate forces and energies, and model complex systems.
  • Computer Science: Expanding algebraic expressions is used in computer algorithms, such as polynomial multiplication and division, and in the design of computer networks.
  • Economics: Expanding algebraic expressions is used to model economic systems, calculate economic indicators, and make predictions about economic trends.

Q: How can I practice expanding algebraic expressions?

A: You can practice expanding algebraic expressions by working through exercises and problems in your textbook or online resources. You can also try expanding expressions on your own, using different variables and coefficients to create new expressions. Additionally, you can use online tools and calculators to help you expand expressions and check your work.