Select The Correct Answer From Each Drop-down Menu.Simplify The Following Polynomial Expression: ( 3 X 2 − X − 7 ) − ( 5 X 2 − 4 X − 2 ) + ( X + 3 ) ( X + 2 \left(3x^2 - X - 7\right) - \left(5x^2 - 4x - 2\right) + (x+3)(x+2 ( 3 X 2 − X − 7 ) − ( 5 X 2 − 4 X − 2 ) + ( X + 3 ) ( X + 2 ]The Polynomial Simplifies To An Expression That Is A $\square$

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Introduction

Polynomial expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying polynomial expressions, focusing on the given expression: (3x2−x−7)−(5x2−4x−2)+(x+3)(x+2)\left(3x^2 - x - 7\right) - \left(5x^2 - 4x - 2\right) + (x+3)(x+2). We will break down the steps involved in simplifying this expression and provide a clear understanding of the process.

Understanding Polynomial Expressions

A polynomial expression is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial expression can be raised to various powers, and the coefficients can be any real number. Polynomial expressions can be classified into different types based on the degree of the highest power of the variable.

Simplifying the Given Expression

To simplify the given expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Evaluate the Expressions Inside the Parentheses

The given expression contains two sets of parentheses: (3x2−x−7)\left(3x^2 - x - 7\right) and (5x2−4x−2)\left(5x^2 - 4x - 2\right). We need to evaluate these expressions separately.

import sympy as sp

x = sp.symbols('x')



expr1 = 3x**2 - x - 7
expr2 = 5x**2 - 4*x - 2


print(expr1)
print(expr2)

Step 2: Subtract the Second Expression from the First Expression

Now that we have evaluated the expressions inside the parentheses, we can subtract the second expression from the first expression.

# Subtract the second expression from the first expression
result = expr1 - expr2

print(result)

Step 3: Multiply the Two Binomials

The expression (x+3)(x+2)(x+3)(x+2) represents the product of two binomials. We can use the distributive property to multiply these binomials.

# Multiply the two binomials
binomial1 = x + 3
binomial2 = x + 2

result = sp.expand(binomial1 * binomial2)


print(result)

Step 4: Add the Results of the Previous Steps

Now that we have multiplied the two binomials, we can add the results of the previous steps.

# Add the results of the previous steps
final_result = result + result

print(final_result)

Conclusion

In this article, we have simplified the given polynomial expression using the order of operations (PEMDAS). We have broken down the steps involved in simplifying the expression and provided a clear understanding of the process. By following these steps, you can simplify any polynomial expression and arrive at the correct solution.

Final Answer

The final answer is: 2x2+9x+8\boxed{2x^2 + 9x + 8}

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Introduction

In our previous article, we explored the process of simplifying polynomial expressions, focusing on the given expression: (3x2−x−7)−(5x2−4x−2)+(x+3)(x+2)\left(3x^2 - x - 7\right) - \left(5x^2 - 4x - 2\right) + (x+3)(x+2). We broke down the steps involved in simplifying this expression and provided a clear understanding of the process. In this article, we will answer some frequently asked questions related to simplifying polynomial expressions.

Q&A

Q: What is the order of operations (PEMDAS) and why is it important?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

  • Parentheses: Evaluate expressions inside parentheses first.
  • Exponents: Evaluate any exponential expressions next.
  • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

The order of operations is important because it ensures that mathematical expressions are evaluated consistently and accurately.

Q: How do I simplify a polynomial expression with multiple terms?

A: To simplify a polynomial expression with multiple terms, follow these steps:

  1. Combine like terms: Combine any terms that have the same variable and exponent.
  2. Simplify each term: Simplify each term by combining any like terms.
  3. Add or subtract terms: Add or subtract the simplified terms to arrive at the final expression.

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

A: A monomial is a single term that consists of a variable and a coefficient. For example, 3x23x^2 is a monomial. A polynomial, on the other hand, is an expression that consists of two or more terms. For example, 3x2+2x+13x^2 + 2x + 1 is a polynomial.

Q: How do I multiply two binomials?

A: To multiply two binomials, use the distributive property to multiply each term in the first binomial by each term in the second binomial. For example, to multiply (x+3)(x+2)(x+3)(x+2), we would multiply each term in the first binomial by each term in the second binomial:

(x+3)(x+2)=x(x+2)+3(x+2)(x+3)(x+2) = x(x+2) + 3(x+2)

=x2+2x+3x+6= x^2 + 2x + 3x + 6

=x2+5x+6= x^2 + 5x + 6

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 2x2+9x+8\boxed{2x^2 + 9x + 8}.

Additional Tips and Resources

  • To simplify polynomial expressions, it's essential to follow the order of operations (PEMDAS).
  • Combine like terms to simplify polynomial expressions.
  • Use the distributive property to multiply binomials.
  • Practice simplifying polynomial expressions to become more comfortable with the process.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying polynomial expressions. We have provided a clear understanding of the process and offered additional tips and resources to help you become more comfortable with simplifying polynomial expressions.

Final Answer

The final answer is: 2x2+9x+8\boxed{2x^2 + 9x + 8}