Simplify The Expression: $(3x-2)^3 - (3x-4)(6x-5) - 45x^2 = 9x^2(3x-5) - 10(x+3) - 2(6x-1) + 6x + 1$

Introduction

In this article, we will simplify the given expression by applying various algebraic techniques. The expression is a combination of multiple terms, and our goal is to simplify it to a single term or a combination of terms that are easier to understand and work with.

The Given Expression

The given expression is:

$$(3x-2)^3 - (3x-4)(6x-5) - 45x^2 = 9x^2(3x-5) - 10(x+3) - 2(6x-1) + 6x + 1$$

Step 1: Expand the Cubic Term

To simplify the expression, we will start by expanding the cubic term $(3x-2)^3$. This can be done using the formula for expanding a cubic term:

$$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$

In this case, $a = 3x$ and $b = 2$. Substituting these values into the formula, we get:

$$(3x-2)^3 = (3x)^3 - 3(3x)^2(2) + 3(3x)(2)^2 - 2^3$$

Simplifying this expression, we get:

$$(3x-2)^3 = 27x^3 - 54x^2 + 36x - 8$$

Step 2: Expand the Product of Two Binomials

Next, we will expand the product of two binomials $(3x-4)(6x-5)$. This can be done using the formula for expanding a product of two binomials:

$$(a-b)(c-d) = ac - ad - bc + bd$$

In this case, $a = 3x$, $b = 4$, $c = 6x$, and $d = 5$. Substituting these values into the formula, we get:

$$(3x-4)(6x-5) = (3x)(6x) - (3x)(5) - (4)(6x) + (4)(5)$$

Simplifying this expression, we get:

$$(3x-4)(6x-5) = 18x^2 - 15x - 24x + 20$$

Combining like terms, we get:

$$(3x-4)(6x-5) = 18x^2 - 39x + 20$$

Step 3: Simplify the Expression

Now that we have expanded the cubic term and the product of two binomials, we can simplify the expression by combining like terms.

The given expression is:

$$(3x-2)^3 - (3x-4)(6x-5) - 45x^2 = 9x^2(3x-5) - 10(x+3) - 2(6x-1) + 6x + 1$$

Substituting the expanded expressions for the cubic term and the product of two binomials, we get:

$$27x^3 - 54x^2 + 36x - 8 - (18x^2 - 39x + 20) - 45x^2 = 9x^2(3x-5) - 10(x+3) - 2(6x-1) + 6x + 1$$

Simplifying this expression, we get:

$$27x^3 - 117x^2 + 75x - 28 = 9x^2(3x-5) - 10(x+3) - 2(6x-1) + 6x + 1$$

Step 4: Expand the Product of Two Binomials

Next, we will expand the product of two binomials $9x^2(3x-5)$. This can be done using the formula for expanding a product of two binomials:

$$(a-b)(c-d) = ac - ad - bc + bd$$

In this case, $a = 9x^2$, $b = 0$, $c = 3x$, and $d = 5$. Substituting these values into the formula, we get:

$$9x^2(3x-5) = (9x^2)(3x) - (9x^2)(5)$$

Simplifying this expression, we get:

$$9x^2(3x-5) = 27x^3 - 45x^2$$

Step 5: Expand the Product of Two Binomials

Next, we will expand the product of two binomials $-10(x+3)$. This can be done using the formula for expanding a product of two binomials:

$$(a-b)(c-d) = ac - ad - bc + bd$$

In this case, $a = -10$, $b = 0$, $c = x$, and $d = 3$. Substituting these values into the formula, we get:

$$-10(x+3) = (-10)(x) - (-10)(3)$$

Simplifying this expression, we get:

$$-10(x+3) = -10x + 30$$

Step 6: Expand the Product of Two Binomials

Next, we will expand the product of two binomials $-2(6x-1)$. This can be done using the formula for expanding a product of two binomials:

$$(a-b)(c-d) = ac - ad - bc + bd$$

In this case, $a = -2$, $b = 0$, $c = 6x$, and $d = 1$. Substituting these values into the formula, we get:

$$-2(6x-1) = (-2)(6x) - (-2)(1)$$

Simplifying this expression, we get:

$$-2(6x-1) = -12x + 2$$

Step 7: Combine Like Terms

Now that we have expanded all the products of two binomials, we can combine like terms to simplify the expression.

The expression is:

$$27x^3 - 117x^2 + 75x - 28 = 27x^3 - 45x^2 - 10x^2 + 75x - 10x + 30 - 12x + 2 + 6x + 1$$

Combining like terms, we get:

$$27x^3 - 117x^2 + 75x - 28 = 27x^3 - 57x^2 + 59x + 33$$

Conclusion

In this article, we simplified the given expression by applying various algebraic techniques. We expanded the cubic term, the product of two binomials, and combined like terms to simplify the expression. The final simplified expression is:

$$27x^3 - 57x^2 + 59x + 33$$

Introduction

In our previous article, we simplified the given expression by applying various algebraic techniques. In this article, we will answer some common questions related to the simplification of the expression.

Q: What is the main goal of simplifying an expression?

A: The main goal of simplifying an expression is to make it easier to understand and work with. Simplifying an expression can help to reduce the complexity of the expression, making it easier to solve problems and make calculations.

Q: What are some common techniques used to simplify expressions?

A: Some common techniques used to simplify expressions include:

• Expanding and combining like terms
• Factoring out common factors
• Canceling out common factors
• Using algebraic identities

Q: How do I know when to use each technique?

A: The choice of technique depends on the specific expression and the goal of simplification. For example, if the expression has multiple terms with the same variable, it may be helpful to combine like terms. If the expression has a common factor, it may be helpful to factor it out.

Q: What is the difference between expanding and combining like terms?

A: Expanding an expression involves multiplying out the terms, while combining like terms involves adding or subtracting terms with the same variable.

Q: How do I expand a product of two binomials?

A: To expand a product of two binomials, you can use the formula:

$$(a-b)(c-d) = ac - ad - bc + bd$$

Q: How do I combine like terms?

A: To combine like terms, you can add or subtract terms with the same variable. For example:

$$2x + 3x = 5x$$

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is important because it can help to:

• Reduce the complexity of the expression
• Make it easier to solve problems and make calculations
• Improve the accuracy of calculations
• Save time and effort

Q: Can I simplify expressions with variables and constants?

A: Yes, you can simplify expressions with variables and constants. The techniques used to simplify expressions with variables and constants are the same as those used to simplify expressions with only variables.

Q: Can I simplify expressions with fractions?

A: Yes, you can simplify expressions with fractions. The techniques used to simplify expressions with fractions are the same as those used to simplify expressions with only variables.

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

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

• Not combining like terms
• Not factoring out common factors
• Not canceling out common factors
• Not using algebraic identities

Conclusion

In this article, we answered some common questions related to the simplification of expressions. We discussed the importance of simplifying expressions, common techniques used to simplify expressions, and common mistakes to avoid. By following these tips and techniques, you can simplify expressions and make it easier to solve problems and make calculations.