Factor The Expression: 12 Y 2 − 32 Y + 21 12y^2 - 32y + 21 12 Y 2 − 32 Y + 21

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Introduction

Factoring an algebraic expression is a fundamental concept in mathematics, and it plays a crucial role in solving equations and inequalities. In this article, we will focus on factoring the given expression $12y^2 - 32y + 21$. Factoring an expression involves expressing it as a product of simpler expressions, called factors. This can be a challenging task, but with the right techniques and strategies, it can be accomplished.

What is Factoring?

Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves finding the factors of the expression, which are the numbers or variables that multiply together to give the original expression. Factoring can be used to simplify expressions, solve equations, and identify the roots of a polynomial.

Types of Factoring

There are several types of factoring, including:

• Greatest Common Factor (GCF) Factoring: This involves finding the greatest common factor of the terms in the expression and factoring it out.
• Difference of Squares Factoring: This involves factoring an expression that is in the form of a difference of squares.
• Sum and Difference of Cubes Factoring: This involves factoring an expression that is in the form of a sum or difference of cubes.
• Grouping Factoring: This involves grouping the terms in the expression and factoring out common factors.

Factoring the Expression $12y^2 - 32y + 21$

To factor the expression $12y^2 - 32y + 21$, we need to find the factors of the expression. We can start by looking for the greatest common factor of the terms in the expression. In this case, the greatest common factor is 1, so we cannot factor out a common factor.

Next, we can try to factor the expression using the difference of squares formula. However, the expression does not fit the form of a difference of squares, so we cannot use this formula.

We can also try to factor the expression using the sum and difference of cubes formula. However, the expression does not fit the form of a sum or difference of cubes, so we cannot use this formula.

Finally, we can try to factor the expression using the grouping method. We can group the terms in the expression as follows:

$12y^2 - 32y + 21 = (12y^2 - 24y) + (-8y + 21)$

We can then factor out common factors from each group:

$(12y^2 - 24y) = 12y(y - 2)$

$(-8y + 21) = -8y + 21$

We can then combine the two groups to get:

$12y^2 - 32y + 21 = 12y(y - 2) - 8y + 21$

We can then factor out a common factor of $-4y + 7$ from the two groups:

$12y^2 - 32y + 21 = (4y - 7)(3y - 3)$

Therefore, the factored form of the expression $12y^2 - 32y + 21$ is $(4y - 7)(3y - 3)$.

Conclusion

Factoring an algebraic expression is a fundamental concept in mathematics, and it plays a crucial role in solving equations and inequalities. In this article, we focused on factoring the expression $12y^2 - 32y + 21$. We used the grouping method to factor the expression, and we found that the factored form of the expression is $(4y - 7)(3y - 3)$. Factoring an expression involves expressing it as a product of simpler expressions, called factors. This can be a challenging task, but with the right techniques and strategies, it can be accomplished.

Tips and Tricks

Here are some tips and tricks for factoring expressions:

• Look for the greatest common factor: The greatest common factor is the largest factor that divides all the terms in the expression.
• Use the difference of squares formula: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$.
• Use the sum and difference of cubes formula: The sum and difference of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
• Use the grouping method: The grouping method involves grouping the terms in the expression and factoring out common factors.

Common Mistakes

Here are some common mistakes to avoid when factoring expressions:

• Not looking for the greatest common factor: Failing to look for the greatest common factor can make it difficult to factor the expression.
• Using the wrong formula: Using the wrong formula can lead to incorrect results.
• Not grouping the terms correctly: Not grouping the terms correctly can make it difficult to factor the expression.

Real-World Applications

Factoring expressions has many real-world applications, including:

• Solving equations and inequalities: Factoring expressions is a key step in solving equations and inequalities.
• Identifying the roots of a polynomial: Factoring expressions can help identify the roots of a polynomial.
• Simplifying expressions: Factoring expressions can help simplify complex expressions.

Conclusion

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Q: What is factoring?

A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions, called factors. It involves finding the factors of the expression, which are the numbers or variables that multiply together to give the original expression.

Q: Why is factoring important?

A: Factoring is an important concept in mathematics because it helps to simplify complex expressions and solve equations and inequalities. It also helps to identify the roots of a polynomial and can be used to solve real-world problems.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

• Greatest Common Factor (GCF) Factoring: This involves finding the greatest common factor of the terms in the expression and factoring it out.
• Difference of Squares Factoring: This involves factoring an expression that is in the form of a difference of squares.
• Sum and Difference of Cubes Factoring: This involves factoring an expression that is in the form of a sum or difference of cubes.
• Grouping Factoring: This involves grouping the terms in the expression and factoring out common factors.

Q: How do I factor an expression?

A: To factor an expression, you need to follow these steps:

1. Look for the greatest common factor: The greatest common factor is the largest factor that divides all the terms in the expression.
2. Use the difference of squares formula: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$.
3. Use the sum and difference of cubes formula: The sum and difference of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
4. Use the grouping method: The grouping method involves grouping the terms in the expression and factoring out common factors.

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

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

• Not looking for the greatest common factor: Failing to look for the greatest common factor can make it difficult to factor the expression.
• Using the wrong formula: Using the wrong formula can lead to incorrect results.
• Not grouping the terms correctly: Not grouping the terms correctly can make it difficult to factor the expression.

Q: How do I know if an expression can be factored?

A: To determine if an expression can be factored, you need to look for the following:

• Common factors: If there are common factors in the expression, you can factor them out.
• Difference of squares: If the expression is in the form of a difference of squares, you can use the difference of squares formula.
• Sum and difference of cubes: If the expression is in the form of a sum or difference of cubes, you can use the sum and difference of cubes formula.
• Grouping: If the expression can be grouped into two or more parts, you can use the grouping method.

Q: Can I factor an expression that has variables with exponents?

A: Yes, you can factor an expression that has variables with exponents. To do this, you need to follow the same steps as factoring an expression with variables without exponents.

Q: Can I factor an expression that has fractions?

A: Yes, you can factor an expression that has fractions. To do this, you need to follow the same steps as factoring an expression without fractions.

Q: How do I check if my factored expression is correct?

A: To check if your factored expression is correct, you need to multiply the factors together and see if you get the original expression. If you do, then your factored expression is correct.

Conclusion

Factoring expressions is an important concept in mathematics that can be used to simplify complex expressions and solve equations and inequalities. By following the steps outlined in this article, you can learn how to factor expressions and become proficient in this skill. Remember to always check your work to ensure that your factored expression is correct.