Factoring by Factoring Out the GCF

Oct 13, 2025 by talynt.org 1 views

Introduction

Factoring polynomials is a fundamental concept in algebra, and it plays a crucial role in solving equations and inequalities. In this article, we will explore the completely factored form of the given polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$. We will examine each option and determine which one is the correct completely factored form.

Option A: $$\left(x^4 + 2x\right)(3x - 7)$$

To determine if this is the correct completely factored form, we need to multiply the two binomials and see if we get the original polynomial.

\left(x^4 + 2x\right)(3x - 7) = 3x^5 - 7x^4 + 6x^2 - 14x

However, when we multiply the two binomials, we get an additional term $$6x^3$$, which is not present in the original polynomial. Therefore, option A is not the correct completely factored form.

Option B: $$x^4(3x - 7)(2x - 1)$$

To determine if this is the correct completely factored form, we need to multiply the three binomials and see if we get the original polynomial.

x^4(3x - 7)(2x - 1) = 3x^5 - 7x^4 + 6x^2 - 14x

When we multiply the three binomials, we get the original polynomial. Therefore, option B is the correct completely factored form.

Option C: $$x\left(x^3 + 2\right)(3x - 7)$$

To determine if this is the correct completely factored form, we need to multiply the three binomials and see if we get the original polynomial.

x\left(x^3 + 2\right)(3x - 7) = 3x^5 - 7x^4 + 6x^2 - 14x

However, when we multiply the three binomials, we get an additional term $$6x^4$$, which is not present in the original polynomial. Therefore, option C is not the correct completely factored form.

Option D: $$x\left(3x^4 - 7x^3 + 6x - 14\right)$$

To determine if this is the correct completely factored form, we need to multiply the two binomials and see if we get the original polynomial.

x\left(3x^4 - 7x^3 + 6x - 14\right) = 3x^5 - 7x^4 + 6x^2 - 14x

However, when we multiply the two binomials, we get an additional term $$-14x^2$$, which is not present in the original polynomial. Therefore, option D is not the correct completely factored form.

Conclusion

In conclusion, the completely factored form of the given polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$ is $$x^4(3x - 7)(2x - 1)$$. This is the correct completely factored form, and it can be verified by multiplying the three binomials and checking if we get the original polynomial.

Step-by-Step Solution

To factor the given polynomial, we can follow these steps:

Factor out the greatest common factor (GCF) of the polynomial, which is $$x$$.
Factor the remaining polynomial $$3x^4 - 7x^3 + 6x - 14$$.
Factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by grouping terms.
Factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by factoring out the GCF of each group of terms.
Factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by factoring out the GCF of each group of terms and then factoring the remaining polynomial.

Factoring by Grouping

To factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by grouping, we can group the terms as follows:

3x^4 - 7x^3 + 6x - 14 = (3x^4 - 7x^3) + (6x - 14)

We can then factor out the GCF of each group of terms:

(3x^4 - 7x^3) + (6x - 14) = x^3(3x - 7) + 2(3x - 7)

We can then factor out the GCF of the two groups of terms:

x^3(3x - 7) + 2(3x - 7) = (3x - 7)(x^3 + 2)

Factoring by Factoring Out the GCF

To factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by factoring out the GCF, we can factor out the GCF of each group of terms:

3x^4 - 7x^3 + 6x - 14 = x^3(3x - 7) + 2(3x - 7)

We can then factor out the GCF of the two groups of terms:

x^3(3x - 7) + 2(3x - 7) = (3x - 7)(x^3 + 2)

Factoring the Remaining Polynomial

To factor the polynomial $$x^3 + 2$$, we can factor it as follows:

x^3 + 2 = (x + \sqrt{2})(x^2 - \sqrt{2}x + 2)

Conclusion

Final Answer

The final answer is: $\boxed{x^4(3x - 7)(2x - 1)}$

Introduction

In our previous article, we explored the completely factored form of the given polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$. We determined that the correct completely factored form is $$x^4(3x - 7)(2x - 1)$$. In this article, we will answer some frequently asked questions about the completely factored form of the given polynomial.

Q: What is the greatest common factor (GCF) of the polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$?

A: The GCF of the polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$ is $$x$$.

Q: How do I factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$?

A: To factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$, you can group the terms and factor out the GCF of each group of terms. You can also factor the polynomial by factoring out the GCF of each group of terms and then factoring the remaining polynomial.

Q: What is the completely factored form of the polynomial $$3x^4 - 7x^3 + 6x - 14$$?

A: The completely factored form of the polynomial $$3x^4 - 7x^3 + 6x - 14$$ is $$(3x - 7)(x^3 + 2)$$.

Q: How do I factor the polynomial $$x^3 + 2$$?

A: To factor the polynomial $$x^3 + 2$$, you can factor it as follows:

x^3 + 2 = (x + \sqrt{2})(x^2 - \sqrt{2}x + 2)

Q: What is the completely factored form of the polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$?

A: The completely factored form of the polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$ is $$x^4(3x - 7)(2x - 1)$$.

Q: How do I verify the completely factored form of the polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$?

A: To verify the completely factored form of the polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$, you can multiply the three binomials and check if you get the original polynomial.

Q: What is the final answer to the problem?

A: The final answer to the problem is $$x^4(3x - 7)(2x - 1)$$.

Conclusion

In conclusion, we have answered some frequently asked questions about the completely factored form of the given polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$. We have also provided some additional information and tips for factoring polynomials.

Step-by-Step Solution

To factor the given polynomial, you can follow these steps:

Factor out the greatest common factor (GCF) of the polynomial, which is $$x$$.
Factor the remaining polynomial $$3x^4 - 7x^3 + 6x - 14$$.
Factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by grouping terms.
Factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by factoring out the GCF of each group of terms.
Factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by factoring out the GCF of each group of terms and then factoring the remaining polynomial.

Factoring by Grouping

To factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by grouping, you can group the terms as follows:

3x^4 - 7x^3 + 6x - 14 = (3x^4 - 7x^3) + (6x - 14)

You can then factor out the GCF of each group of terms:

(3x^4 - 7x^3) + (6x - 14) = x^3(3x - 7) + 2(3x - 7)

You can then factor out the GCF of the two groups of terms:

x^3(3x - 7) + 2(3x - 7) = (3x - 7)(x^3 + 2)

Factoring by Factoring Out the GCF

To factor the polynomial $$3x^4 - 7x^3 + 6x - 14$$ by factoring out the GCF, you can factor out the GCF of each group of terms:

3x^4 - 7x^3 + 6x - 14 = x^3(3x - 7) + 2(3x - 7)

You can then factor out the GCF of the two groups of terms:

x^3(3x - 7) + 2(3x - 7) = (3x - 7)(x^3 + 2)

Factoring the Remaining Polynomial

To factor the polynomial $$x^3 + 2$$, you can factor it as follows:

x^3 + 2 = (x + \sqrt{2})(x^2 - \sqrt{2}x + 2)

Conclusion

In conclusion, we have provided a step-by-step solution to the problem and answered some frequently asked questions about the completely factored form of the given polynomial $$3x^5 - 7x^4 + 6x^2 - 14x$$.

Final Answer

The final answer is: $\boxed{x^4(3x - 7)(2x - 1)}$