Simplify The Expression: 1 − 3 X 2 ( 4 X 2 − 2 X + 7 1 - 3x^2(4x^2 - 2x + 7 1 − 3 X 2 ( 4 X 2 − 2 X + 7 ]

Mar 12, 2025 by ADMIN 106 views

Simplify the Expression: $1 - 3x^2(4x^2 - 2x + 7)$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of algebra. In this article, we will simplify the given expression $1 - 3x^2(4x^2 - 2x + 7)$ using various algebraic techniques. We will break down the expression into smaller parts, apply the distributive property, and combine like terms to simplify the expression.

Understanding the Expression

The given expression is a quadratic expression in the variable $x$ . It consists of a constant term, a linear term, and a quadratic term. The expression can be written as:

1 - 3x^2(4x^2 - 2x + 7)

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , we have:

a(b + c) = ab + ac

Using the distributive property, we can rewrite the expression as:

1 - 3x^2(4x^2) + 1 - 3x^2(-2x) + 1 - 3x^2(7)

Simplifying the Expression

Now, we can simplify the expression by combining like terms. We have three terms with the same variable $x^2$ , and we can combine them as follows:

1 - 12x^4 + 6x^3 - 21x^2

Combining Like Terms

We can further simplify the expression by combining like terms. We have two terms with the same variable $x^3$ , and we can combine them as follows:

1 - 12x^4 + 6x^3 - 21x^2

Final Simplified Expression

The final simplified expression is:

1 - 12x^4 + 6x^3 - 21x^2

Conclusion

In this article, we simplified the given expression $1 - 3x^2(4x^2 - 2x + 7)$ using the distributive property and combining like terms. We broke down the expression into smaller parts, applied the distributive property, and combined like terms to simplify the expression. The final simplified expression is $1 - 12x^4 + 6x^3 - 21x^2$ .

Tips and Tricks

When simplifying algebraic expressions, it is essential to use the distributive property to break down the expression into smaller parts.
Combining like terms is a crucial step in simplifying algebraic expressions.
Always check your work by plugging in values for the variable to ensure that the simplified expression is correct.

Common Mistakes

Failing to use the distributive property can lead to incorrect simplification of algebraic expressions.
Not combining like terms can result in a more complex expression than necessary.
Not checking your work can lead to errors in the simplified expression.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications in various fields, including:

Physics: Simplifying algebraic expressions is essential in solving problems related to motion, energy, and momentum.
Engineering: Simplifying algebraic expressions is crucial in designing and analyzing complex systems, such as bridges, buildings, and electronic circuits.
Computer Science: Simplifying algebraic expressions is essential in programming and algorithm design.

Final Thoughts

Simplifying algebraic expressions is a fundamental skill in mathematics, and it requires a deep understanding of the rules of algebra. By using the distributive property and combining like terms, we can simplify complex expressions and arrive at a final simplified expression. Remember to always check your work and use the distributive property to break down the expression into smaller parts. With practice and patience, you will become proficient in simplifying algebraic expressions and apply them to real-world problems.

Introduction

In our previous article, we simplified the given expression $1 - 3x^2(4x^2 - 2x + 7)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , we have:

a(b + c) = ab + ac

This property allows us to break down a complex expression into smaller parts and simplify it.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

Use the distributive property to break down the expression into smaller parts.
Combine like terms by adding or subtracting the coefficients of the same variable.
Check your work by plugging in values for the variable to ensure that the simplified expression is correct.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $5x^2$ are like terms because they have the same variable $x$ and exponent $2$ .

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the same variable. For example, $2x^2 + 5x^2 = 7x^2$ .

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, such as $x$ or $y$ . A constant is a value that does not change, such as $2$ or $5$ .

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, use the distributive property to break down the expression into smaller parts, and then combine like terms.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

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

Failing to use the distributive property
Not combining like terms
Not checking your work
Making errors in the order of operations

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

A: To check your work, plug in values for the variable to ensure that the simplified expression is correct. You can also use a calculator or a computer algebra system to check your work.