What Is The Fully Factored Form Of $27x^3 + 7x^2$?Use The Caret Symbol (^) To Denote Exponents In Your Answer. For Example, If Your Answer Is $17x^2 + 9$, You Would Type It As $x^{\wedge}3(17x^{\wedge}2 + 9)$ With No

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What is the Fully Factored Form of 27x^3 + 7x^2?

In algebra, factoring is a process of expressing an algebraic expression as a product of simpler expressions. The fully factored form of an expression is the expression that is factored into the simplest possible form. In this article, we will explore the fully factored form of the expression 27x^3 + 7x^2.

The given expression is 27x^3 + 7x^2. To factor this expression, we need to find the greatest common factor (GCF) of the two terms. The GCF of 27x^3 and 7x^2 is 7x^2.

To factor out the GCF, we need to divide each term by the GCF. In this case, we divide 27x^3 by 7x^2 and 7x^2 by 7x^2.

27x^3 ÷ 7x^2 = 3x 7x^2 ÷ 7x^2 = 1

So, the expression can be written as:

27x^3 + 7x^2 = 7x^2(3x + 1)

To check if the factored form is correct, we need to multiply the factors together and see if we get the original expression.

7x^2(3x + 1) = 7x^2(3x) + 7x^2(1) = 21x^3 + 7x^2

As we can see, the factored form is correct.

In conclusion, the fully factored form of 27x^3 + 7x^2 is 7x^2(3x + 1). This is the simplest form of the expression, and it can be obtained by factoring out the greatest common factor (GCF) of the two terms.

The fully factored form of an expression can be useful in solving equations and inequalities. For example, if we have an equation like 27x^3 + 7x^2 = 0, we can factor the left-hand side of the equation as 7x^2(3x + 1) = 0. This tells us that either 7x^2 = 0 or 3x + 1 = 0. We can then solve each of these equations separately to find the solutions to the original equation.

  • When factoring an expression, always look for the greatest common factor (GCF) of the terms.
  • Use the distributive property to multiply the factors together and check if the factored form is correct.
  • The fully factored form of an expression can be useful in solving equations and inequalities.
  • Not factoring out the greatest common factor (GCF) of the terms.
  • Not using the distributive property to multiply the factors together and check if the factored form is correct.
  • Not simplifying the expression as much as possible.

Q: What is the fully factored form of 27x^3 + 7x^2?

A: The fully factored form of 27x^3 + 7x^2 is 7x^2(3x + 1).

Q: How do I factor out the greatest common factor (GCF) of the terms?

A: To factor out the GCF, you need to divide each term by the GCF. In this case, you divide 27x^3 by 7x^2 and 7x^2 by 7x^2.

Q: What is the distributive property, and how do I use it to multiply the factors together?

A: The distributive property is a rule that states that a(b + c) = ab + ac. To use it to multiply the factors together, you need to multiply each term in the first factor by each term in the second factor.

Q: How do I check if the factored form is correct?

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

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

A: Some common mistakes to avoid when factoring an expression include not factoring out the greatest common factor (GCF) of the terms, not using the distributive property to multiply the factors together, and not simplifying the expression as much as possible.

Q: Why is the fully factored form of an expression important?

A: The fully factored form of an expression is important because it can be useful in solving equations and inequalities. It can also help you to simplify complex expressions and make them easier to work with.

Q: Can you give an example of how to use the fully factored form of an expression to solve an equation?

A: Yes, here is an example. Suppose we have an equation like 27x^3 + 7x^2 = 0. We can factor the left-hand side of the equation as 7x^2(3x + 1) = 0. This tells us that either 7x^2 = 0 or 3x + 1 = 0. We can then solve each of these equations separately to find the solutions to the original equation.

Q: What are some tips and tricks for factoring expressions?

A: Some tips and tricks for factoring expressions include looking for the greatest common factor (GCF) of the terms, using the distributive property to multiply the factors together, and simplifying the expression as much as possible.

Q: Can you give some examples of expressions that can be factored using the greatest common factor (GCF)?

A: Yes, here are some examples. Suppose we have an expression like 12x^2 + 18x. We can factor out the greatest common factor (GCF) of 6x to get 6x(2x + 3). Another example is the expression 24x^3 + 12x^2. We can factor out the greatest common factor (GCF) of 12x^2 to get 12x^2(2x + 1).

In conclusion, the fully factored form of 27x^3 + 7x^2 is 7x^2(3x + 1). This is the simplest form of the expression, and it can be obtained by factoring out the greatest common factor (GCF) of the two terms. The fully factored form of an expression can be useful in solving equations and inequalities, and it is an important concept in algebra.