Match The Polynomial Expression On The Left With The Simplified Version On The Right.$[ \begin{array}{ll} (2x - 3)^2 & \quad 4x^2 - 12x + 9 \ (2x - 3)(2x + 3) & \quad 4x^2 - 9 \ & \quad 4x^2 + 9 \ & \quad 4x - 6

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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 problem of matching the polynomial expression on the left with the simplified version on the right.

What are Polynomial Expressions?

A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form of a sum of terms, where each term is a product of a coefficient and one or more variables raised to a non-negative integer power.

The Given Problem

We are given three polynomial expressions on the left and four simplified versions on the right. Our task is to match each polynomial expression on the left with the simplified version on the right.

Step 1: Expand the First Polynomial Expression

The first polynomial expression is (2x−3)2(2x - 3)^2. To simplify this expression, we need to expand it using the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.

import sympy as sp

x = sp.symbols('x')



expr1 = (2*x - 3)**2



expanded_expr1 = sp.expand(expr1)


print(expanded_expr1)

The expanded expression is 4x2−12x+94x^2 - 12x + 9. This matches the first simplified version on the right.

Step 2: Multiply the Second Polynomial Expression

The second polynomial expression is (2x−3)(2x+3)(2x - 3)(2x + 3). To simplify this expression, we need to multiply the two binomials using the distributive property.

# Define the polynomial expression
expr2 = (2*x - 3)*(2*x + 3)


multiplied_expr2 = sp.expand(expr2)


print(multiplied_expr2)

The multiplied expression is 4x2−94x^2 - 9. This matches the second simplified version on the right.

Step 3: Simplify the Third Polynomial Expression

The third polynomial expression is 4x2+94x^2 + 9. This expression is already simplified, so we don't need to do anything.

Step 4: Simplify the Fourth Polynomial Expression

The fourth polynomial expression is 4x−64x - 6. This expression is already simplified, so we don't need to do anything.

Conclusion

In this article, we have explored the process of simplifying polynomial expressions, focusing on the given problem of matching the polynomial expression on the left with the simplified version on the right. We have used the distributive property and the formula for expanding a squared binomial to simplify the expressions. The final answer is that the first polynomial expression matches the first simplified version, the second polynomial expression matches the second simplified version, and the third and fourth polynomial expressions are already simplified.

Discussion

This problem is a great example of how to simplify polynomial expressions using different techniques. The distributive property and the formula for expanding a squared binomial are two powerful tools that can be used to simplify polynomial expressions. By practicing these techniques, you can become more confident and proficient in simplifying polynomial expressions.

Additional Resources

If you want to learn more about simplifying polynomial expressions, here are some additional resources that you may find helpful:

  • Khan Academy: Simplifying Polynomial Expressions
  • Mathway: Simplifying Polynomial Expressions
  • Wolfram Alpha: Simplifying Polynomial Expressions

Final Answer

Q: What is a polynomial expression?

A: A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form of a sum of terms, where each term is a product of a coefficient and one or more variables raised to a non-negative integer power.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you can use the following techniques:

  • Expand a squared binomial using the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
  • Multiply two binomials using the distributive property
  • Combine like terms by adding or subtracting coefficients of the same variable

Q: What is the distributive property?

A: The distributive property is a rule that states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property can be used to multiply two binomials.

Q: How do I expand a squared binomial?

A: To expand a squared binomial, you can use the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2. For example, if you want to expand (2x−3)2(2x - 3)^2, you would use the formula as follows:

(2x−3)2=(2x)2−2(2x)(3)+32(2x - 3)^2 = (2x)^2 - 2(2x)(3) + 3^2 =4x2−12x+9= 4x^2 - 12x + 9

Q: What is the difference between a polynomial expression and an algebraic expression?

A: A polynomial expression is a specific type of algebraic expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include any combination of variables, coefficients, and mathematical operations.

Q: Can I simplify a polynomial expression with negative coefficients?

A: Yes, you can simplify a polynomial expression with negative coefficients. To do this, you can use the same techniques as before, such as expanding a squared binomial or multiplying two binomials. For example, if you want to simplify −(2x−3)2-(2x - 3)^2, you would use the formula as follows:

−(2x−3)2=−((2x)2−2(2x)(3)+32)-(2x - 3)^2 = -((2x)^2 - 2(2x)(3) + 3^2) =−4x2+12x−9= -4x^2 + 12x - 9

Q: How do I know when a polynomial expression is already simplified?

A: A polynomial expression is already simplified if it cannot be further simplified using the techniques mentioned above. For example, if you have the expression 4x2+94x^2 + 9, it is already simplified because there are no like terms to combine.

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

A: Yes, you can use a calculator to simplify a polynomial expression. Many calculators have a built-in function for simplifying polynomial expressions. However, it's always a good idea to double-check your work by hand to make sure that the calculator is giving you the correct answer.

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

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

  • Forgetting to combine like terms
  • Not using the distributive property when multiplying two binomials
  • Not using the formula for expanding a squared binomial
  • Not checking your work by hand to make sure that the calculator is giving you the correct answer.