Multiply The Expressions:${ \frac{3x^2 + 2x - 21}{-2x^2 - 2x + 12} \cdot \frac{2x^2 + 25x + 63}{6x^2 + 7x - 49} }$If A = 1 A=1 A = 1 , Find The Values Of B , C B, C B , C , And D D D That Make The Given Expression Equivalent To The

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. One of the most common methods of simplifying expressions is by multiplying them together. However, when dealing with algebraic fractions, multiplying them can be a bit more challenging. In this article, we will explore how to multiply algebraic fractions and simplify the resulting expression.

Multiplying Algebraic Fractions

To multiply algebraic fractions, we need to follow the same rules as multiplying regular fractions. We multiply the numerators together and the denominators together. However, when dealing with algebraic fractions, we need to be careful with the signs and the order of operations.

Example

Let's consider the following expression:

{ \frac{3x^2 + 2x - 21}{-2x^2 - 2x + 12} \cdot \frac{2x^2 + 25x + 63}{6x^2 + 7x - 49} \}

To multiply these fractions, we need to multiply the numerators together and the denominators together.

{ \frac{(3x^2 + 2x - 21)(2x^2 + 25x + 63)}{(-2x^2 - 2x + 12)(6x^2 + 7x - 49)} \}

Expanding the Numerator and Denominator

To simplify the expression, we need to expand the numerator and denominator.

Expanding the Numerator

To expand the numerator, we need to multiply each term in the first fraction by each term in the second fraction.

{ (3x^2 + 2x - 21)(2x^2 + 25x + 63) = 6x^4 + 75x^3 + 189x^2 + 4x^3 + 50x^2 + 126x - 42x^2 - 525x - 1383 \}

Combining like terms, we get:

{ 6x^4 + 79x^3 + 101x^2 - 399x - 1383 \}

Expanding the Denominator

To expand the denominator, we need to multiply each term in the first fraction by each term in the second fraction.

{ (-2x^2 - 2x + 12)(6x^2 + 7x - 49) = -12x^4 - 14x^3 + 98x^2 - 12x^3 - 14x^2 + 98x + 72x^2 + 84x - 588 \}

Combining like terms, we get:

{ -12x^4 - 24x^3 + 168x^2 + 182x - 588 \}

Simplifying the Expression

Now that we have expanded the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator.

{ \frac{6x^4 + 79x^3 + 101x^2 - 399x - 1383}{-12x^4 - 24x^3 + 168x^2 + 182x - 588} \}

Factoring the Numerator and Denominator

To simplify the expression further, we can factor the numerator and denominator.

Factoring the Numerator

To factor the numerator, we need to find the greatest common factor (GCF) of the terms.

{ 6x^4 + 79x^3 + 101x^2 - 399x - 1383 = (3x^2 + 11x + 21)(2x^2 + 7x - 33) \}

Factoring the Denominator

To factor the denominator, we need to find the greatest common factor (GCF) of the terms.

{ -12x^4 - 24x^3 + 168x^2 + 182x - 588 = (-4x^2 - 4x + 12)(3x^2 + 7x - 49) \}

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel common factors.

{ \frac{(3x^2 + 11x + 21)(2x^2 + 7x - 33)}{(-4x^2 - 4x + 12)(3x^2 + 7x - 49)} = \frac{(3x^2 + 11x + 21)(2x^2 + 7x - 33)}{(-4x^2 - 4x + 12)(3x^2 + 7x - 49)} \}

Conclusion

In conclusion, multiplying algebraic fractions can be a bit more challenging than multiplying regular fractions. However, by following the same rules and being careful with the signs and the order of operations, we can simplify the resulting expression. In this article, we have explored how to multiply algebraic fractions and simplify the resulting expression.

Discussion

If a=1a=1, find the values of b,cb, c, and dd that make the given expression equivalent to the expression:

{ \frac{(3x^2 + 11x + 21)(2x^2 + 7x - 33)}{(-4x^2 - 4x + 12)(3x^2 + 7x - 49)} \}

To find the values of b,cb, c, and dd, we need to set up an equation and solve for the unknowns.

Setting Up the Equation

Let's set up an equation using the given expression and the expression we want to find.

{ \frac{(3x^2 + 11x + 21)(2x^2 + 7x - 33)}{(-4x^2 - 4x + 12)(3x^2 + 7x - 49)} = \frac{bx^2 + cx + d}{ax^2 + bx + c} \}

Solving for the Unknowns

To solve for the unknowns, we need to equate the coefficients of the terms on both sides of the equation.

{ \frac{(3x^2 + 11x + 21)(2x^2 + 7x - 33)}{(-4x^2 - 4x + 12)(3x^2 + 7x - 49)} = \frac{bx^2 + cx + d}{ax^2 + bx + c} \}

Equating the coefficients of the x2x^2 terms, we get:

{ \frac{6x^4 + 79x^3 + 101x^2 - 399x - 1383}{-12x^4 - 24x^3 + 168x^2 + 182x - 588} = \frac{bx^2 + cx + d}{ax^2 + bx + c} \}

Equating the coefficients of the xx terms, we get:

{ \frac{6x^4 + 79x^3 + 101x^2 - 399x - 1383}{-12x^4 - 24x^3 + 168x^2 + 182x - 588} = \frac{bx^2 + cx + d}{ax^2 + bx + c} \}

Equating the constant terms, we get:

{ \frac{6x^4 + 79x^3 + 101x^2 - 399x - 1383}{-12x^4 - 24x^3 + 168x^2 + 182x - 588} = \frac{bx^2 + cx + d}{ax^2 + bx + c} \}

Solving the system of equations, we get:

{ b = 3, c = 11, d = 21 \}

Therefore, the values of b,cb, c, and dd that make the given expression equivalent to the expression are b=3,c=11,d=21b=3, c=11, d=21.

Final Answer

The final answer is: 3,11,21\boxed{3, 11, 21}

Introduction

In our previous article, we explored how to multiply algebraic fractions and simplify the resulting expression. However, we know that there are still many questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions about multiplying algebraic fractions.

Q1: What is the difference between multiplying algebraic fractions and multiplying regular fractions?

A1: The main difference between multiplying algebraic fractions and multiplying regular fractions is that algebraic fractions involve variables and exponents, whereas regular fractions do not. When multiplying algebraic fractions, we need to be careful with the signs and the order of operations.

Q2: How do I multiply algebraic fractions with different variables?

A2: When multiplying algebraic fractions with different variables, we need to use the distributive property to multiply each term in the first fraction by each term in the second fraction. We also need to be careful with the signs and the order of operations.

Q3: Can I cancel common factors when multiplying algebraic fractions?

A3: Yes, we can cancel common factors when multiplying algebraic fractions. However, we need to make sure that the common factors are not zero and that the resulting expression is still valid.

Q4: How do I simplify the resulting expression after multiplying algebraic fractions?

A4: To simplify the resulting expression, we need to combine like terms and factor out any common factors. We also need to be careful with the signs and the order of operations.

Q5: Can I use algebraic fractions to solve equations and inequalities?

A5: Yes, we can use algebraic fractions to solve equations and inequalities. However, we need to be careful with the signs and the order of operations, and we need to make sure that the resulting expression is still valid.

Q6: How do I find the values of b, c, and d that make the given expression equivalent to the expression?

A6: To find the values of b, c, and d, we need to set up an equation and solve for the unknowns. We can use the distributive property to multiply each term in the first fraction by each term in the second fraction, and we can combine like terms to simplify the resulting expression.

Q7: Can I use algebraic fractions to solve systems of equations?

A7: Yes, we can use algebraic fractions to solve systems of equations. However, we need to be careful with the signs and the order of operations, and we need to make sure that the resulting expression is still valid.

Q8: How do I check my work when multiplying algebraic fractions?

A8: To check our work, we need to make sure that the resulting expression is still valid and that the signs and the order of operations are correct. We can also use algebraic manipulations to simplify the expression and check our work.

Q9: Can I use algebraic fractions to solve word problems?

A9: Yes, we can use algebraic fractions to solve word problems. However, we need to be careful with the signs and the order of operations, and we need to make sure that the resulting expression is still valid.

Q10: How do I apply algebraic fractions to real-world problems?

A10: To apply algebraic fractions to real-world problems, we need to use algebraic manipulations to simplify the expression and check our work. We can also use algebraic fractions to solve systems of equations and inequalities, and to find the values of b, c, and d that make the given expression equivalent to the expression.

Conclusion

In conclusion, multiplying algebraic fractions can be a bit more challenging than multiplying regular fractions. However, by following the same rules and being careful with the signs and the order of operations, we can simplify the resulting expression and solve equations and inequalities. We hope that this Q&A article has helped to address some of the most frequently asked questions about multiplying algebraic fractions.

Final Answer

The final answer is: 3,11,21\boxed{3, 11, 21}