Simplify The Expression: 12 K 2 − 111 K + 27 12k^2 - 111k + 27 12 K 2 − 111 K + 27

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. It involves rewriting an expression in a simpler form, often by combining like terms or factoring out common factors. In this article, we will simplify the expression $12k^2 - 111k + 27$ using various algebraic techniques.

Understanding the Expression

The given expression is a quadratic expression in the form of $ax^2 + bx + c$, where $a = 12$, $b = -111$, and $c = 27$. To simplify this expression, we need to examine its structure and identify any patterns or relationships between the terms.

Factoring Out Common Factors

One way to simplify the expression is to factor out common factors from the terms. In this case, we can factor out the greatest common factor (GCF) of the coefficients, which is 3.

$12k^2 - 111k + 27$
= $3(4k^2 - 37k + 9)$

By factoring out 3, we have simplified the expression and made it easier to work with.

Grouping Terms

Another technique for simplifying the expression is to group the terms into pairs. This can help us identify any patterns or relationships between the terms.

$12k^2 - 111k + 27$
= $(12k^2 - 108k) - (3k + 27)$

By grouping the terms, we can see that the first pair of terms has a common factor of $12k$, while the second pair of terms has a common factor of 3.

Factoring the Expression

Now that we have grouped the terms, we can try to factor the expression further. We can start by factoring the first pair of terms, which has a common factor of $12k$.

$(12k^2 - 108k) - (3k + 27)$
= $12k(k - 9) - 3(k + 9)$

By factoring the first pair of terms, we have simplified the expression and made it easier to work with.

Simplifying the Expression

Now that we have factored the expression, we can simplify it further by combining like terms. We can start by combining the terms with the same variable, which is $k$.

$12k(k - 9) - 3(k + 9)$
= $12k^2 - 108k - 3k - 27$
= $12k^2 - 111k - 27$

By combining the like terms, we have simplified the expression and obtained the final result.

Conclusion

In this article, we simplified the expression $12k^2 - 111k + 27$ using various algebraic techniques. We factored out common factors, grouped terms, and factored the expression to obtain the final result. By simplifying the expression, we have made it easier to work with and have obtained a more manageable form.

Final Answer

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Introduction

In our previous article, we simplified the expression $12k^2 - 111k + 27$ using various algebraic techniques. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of the expression.

Q&A

Q: What is the greatest common factor (GCF) of the coefficients in the expression $12k^2 - 111k + 27$?

A: The GCF of the coefficients is 3.

Q: How do you factor out the GCF from the expression $12k^2 - 111k + 27$?

A: To factor out the GCF, we divide each term by the GCF. In this case, we divide each term by 3.

$12k^2 - 111k + 27$
= $3(4k^2 - 37k + 9)$

Q: What is the difference between factoring out the GCF and factoring the expression?

A: Factoring out the GCF involves dividing each term by the GCF, while factoring the expression involves finding the product of two or more expressions that when multiplied together give the original expression.

Q: How do you group the terms in the expression $12k^2 - 111k + 27$?

A: To group the terms, we pair the terms that have a common factor. In this case, we pair the first two terms and the last two terms.

$12k^2 - 111k + 27$
= $(12k^2 - 108k) - (3k + 27)$

Q: What is the common factor of the first pair of terms in the expression $12k^2 - 111k + 27$?

A: The common factor of the first pair of terms is $12k$.

Q: How do you factor the expression $12k^2 - 111k + 27$ further?

A: To factor the expression further, we can use the factored form of the first pair of terms and try to factor the second pair of terms.

$(12k^2 - 108k) - (3k + 27)$
= $12k(k - 9) - 3(k + 9)$

Q: What is the final simplified form of the expression $12k^2 - 111k + 27$?

A: The final simplified form of the expression is $12k^2 - 111k - 27$.

Q: Why is it important to simplify expressions in algebra?

A: Simplifying expressions in algebra is important because it helps us to:

• Make the expression easier to work with
• Identify any patterns or relationships between the terms
• Solve equations and inequalities more efficiently
• Understand the underlying structure of the expression

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the simplification of the expression $12k^2 - 111k + 27$. We covered topics such as factoring out the GCF, grouping terms, and factoring the expression. By simplifying the expression, we have made it easier to work with and have obtained a more manageable form.

Final Answer

The final answer is: $\boxed{12k^2 - 111k - 27}$