Understanding Factoring Polynomials

Oct 13, 2025 by magicalceylontours.com 136 views

Understanding the Problem

When dealing with polynomial expressions, factoring is an essential concept in mathematics. It involves expressing a polynomial as a product of simpler polynomials, known as factors. In this problem, we are given a polynomial expression $$x^3+343$$ and asked to determine which of the given options is a factor of this expression.

Analyzing the Options

To determine which option is a factor, we need to examine each option carefully. Let's start by analyzing each option one by one.

Option A: $$x-7$$

To check if $$x-7$$ is a factor of $$x^3+343$$, we can use the factor theorem. The factor theorem states that if $$f(a) = 0$$, then $$(x-a)$$ is a factor of $$f(x)$$. In this case, we need to check if $$f(7) = 0$$, where $$f(x) = x^3+343$$.

Substituting $$x = 7$$ into the expression, we get:

f(7) = 7^3 + 343

f(7) = 343 + 343

f(7) = 686

Since $$f(7) \neq 0$$, we can conclude that $$x-7$$ is not a factor of $$x^3+343$$.

Option B: $$x^2-14x+49$$

To check if $$x^2-14x+49$$ is a factor of $$x^3+343$$, we can use polynomial long division or synthetic division. However, we can also use the fact that $$x^3+343$$ can be written as $$(x+7)^3$$.

Expanding $$(x+7)^3$$, we get:

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

= x^3 + 21x^2 + 147x + 343

Comparing this with $$x^3+343$$, we can see that $$x^2-14x+49$$ is not a factor of $$x^3+343$$.

Option C: $$x^2+7x+49$$

To check if $$x^2+7x+49$$ is a factor of $$x^3+343$$, we can use polynomial long division or synthetic division. However, we can also use the fact that $$x^3+343$$ can be written as $$(x+7)^3$$.

Expanding $$(x+7)^3$$, we get:

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

= x^3 + 21x^2 + 147x + 343

Comparing this with $$x^3+343$$, we can see that $$x^2+7x+49$$ is not a factor of $$x^3+343$$.

Option D: $$x+7$$

To check if $$x+7$$ is a factor of $$x^3+343$$, we can use the factor theorem. The factor theorem states that if $$f(a) = 0$$, then $$(x-a)$$ is a factor of $$f(x)$$. In this case, we need to check if $$f(-7) = 0$$, where $$f(x) = x^3+343$$.

Substituting $$x = -7$$ into the expression, we get:

f(-7) = (-7)^3 + 343

f(-7) = -343 + 343

f(-7) = 0

Since $$f(-7) = 0$$, we can conclude that $$x+7$$ is a factor of $$x^3+343$$.

Conclusion

In conclusion, we have analyzed each option carefully and determined that only $$x+7$$ is a factor of $$x^3+343$$. This is because $$x+7$$ satisfies the factor theorem, and the other options do not.

Final Answer

The final answer is: $\boxed{D}$

Understanding Factoring Polynomials

Factoring polynomials is an essential concept in mathematics, and it involves expressing a polynomial as a product of simpler polynomials, known as factors. In the previous article, we discussed how to determine which of the given options is a factor of the polynomial expression $$x^3+343$$. In this article, we will provide a Q&A section to help you better understand factoring polynomials.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, known as factors. For example, if we have a polynomial expression $$x^2+5x+6$$, we can factor it as $$(x+2)(x+3)$$.

Q: How do I determine if a polynomial is factorable?

A: To determine if a polynomial is factorable, we need to check if it can be expressed as a product of simpler polynomials. We can use various methods such as polynomial long division, synthetic division, or the factor theorem to determine if a polynomial is factorable.

Q: What is the factor theorem?

A: The factor theorem states that if $$f(a) = 0$$, then $$(x-a)$$ is a factor of $$f(x)$$. This means that if we substitute a value of $$x$$ into the polynomial expression and get a result of $$0$$, then $$(x-a)$$ is a factor of the polynomial.

Q: How do I use the factor theorem to determine if a polynomial is factorable?

A: To use the factor theorem, we need to substitute a value of $$x$$ into the polynomial expression and check if the result is $$0$$. If the result is $$0$$, then $$(x-a)$$ is a factor of the polynomial.

Q: What are some common methods for factoring polynomials?

A: Some common methods for factoring polynomials include:

Polynomial long division
Synthetic division
The factor theorem
Grouping
Greatest common factor (GCF)

Q: How do I factor a polynomial using polynomial long division?

A: To factor a polynomial using polynomial long division, we need to divide the polynomial by a binomial or a trinomial. We can use the following steps:

Divide the leading term of the polynomial by the leading term of the divisor.
Multiply the entire divisor by the result and subtract the product from the polynomial.
Bring down the next term of the polynomial and repeat the process until we have factored the polynomial completely.

Q: How do I factor a polynomial using synthetic division?

A: To factor a polynomial using synthetic division, we need to divide the polynomial by a binomial or a trinomial. We can use the following steps:

Write the coefficients of the polynomial in a row.
Write the root of the divisor in the first column.
Multiply the root by the first coefficient and write the result below the second coefficient.
Add the second coefficient and the result and write the result below the third coefficient.
Repeat the process until we have factored the polynomial completely.

Q: How do I factor a polynomial using the factor theorem?

A: To factor a polynomial using the factor theorem, we need to substitute a value of $$x$$ into the polynomial expression and check if the result is $$0$$. If the result is $$0$$, then $$(x-a)$$ is a factor of the polynomial.

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

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

Not checking if the polynomial is factorable before attempting to factor it.
Not using the correct method for factoring the polynomial.
Not checking if the factors are correct.
Not simplifying the factors.

Conclusion

In conclusion, factoring polynomials is an essential concept in mathematics, and it involves expressing a polynomial as a product of simpler polynomials, known as factors. We have provided a Q&A section to help you better understand factoring polynomials and avoid common mistakes.

Final Tips

Always check if the polynomial is factorable before attempting to factor it.
Use the correct method for factoring the polynomial.
Check if the factors are correct.
Simplify the factors.

By following these tips, you can become proficient in factoring polynomials and solve complex mathematical problems with ease.