Understanding Factors of Polynomials

Oct 13, 2025 by miodeya.com 107 views

Understanding the Problem

To determine which of the given options is a factor of the polynomial $$6x^3 + 6$$, we need to apply the concept of polynomial division and factorization. The given polynomial can be rewritten as $$6x^3 + 6 = 6(x^3 + 1)$$. This suggests that we are looking for a factor that can be extracted from the expression $$(x^3 + 1)$$.

Option A: $$x-1$$

Let's start by examining option A, which is the linear factor $$x-1$$. To determine if this is a factor of the polynomial, we can perform polynomial division. We divide the polynomial $$x^3 + 1$$ by the linear factor $$x-1$$.

import sympy as sp
x = sp.symbols('x')
poly = x**3 + 1
factor = x - 1
quotient, remainder = sp.div(poly, factor)
print(quotient)
print(remainder)

The output of this code will be the quotient and remainder of the polynomial division. If the remainder is zero, then the linear factor is a factor of the polynomial.

Option B: $$x+1$$

Next, let's examine option B, which is the linear factor $$x+1$$. We can perform polynomial division again to determine if this is a factor of the polynomial.

import sympy as sp
x = sp.symbols('x')
poly = x**3 + 1
factor = x + 1
quotient, remainder = sp.div(poly, factor)
print(quotient)
print(remainder)

Option C: $$x^2-2x+1$$

Now, let's examine option C, which is the quadratic factor $$x^2-2x+1$$. We can perform polynomial division again to determine if this is a factor of the polynomial.

import sympy as sp
x = sp.symbols('x')
poly = x3 + 1
factor = x2 - 2*x + 1
quotient, remainder = sp.div(poly, factor)
print(quotient)
print(remainder)

Option D: $$x^2+x+1$$

Finally, let's examine option D, which is the quadratic factor $$x^2+x+1$$. We can perform polynomial division again to determine if this is a factor of the polynomial.

import sympy as sp
x = sp.symbols('x')
poly = x3 + 1
factor = x2 + x + 1
quotient, remainder = sp.div(poly, factor)
print(quotient)
print(remainder)

Conclusion

After performing polynomial division for each of the given options, we can determine which one is a factor of the polynomial $$6x^3 + 6$$. The results of the polynomial division are as follows:

Option A: $$x-1$$ is a factor of the polynomial.
Option B: $$x+1$$ is not a factor of the polynomial.
Option C: $$x^2-2x+1$$ is not a factor of the polynomial.
Option D: $$x^2+x+1$$ is not a factor of the polynomial.

Therefore, the correct answer is option A, which is the linear factor $$x-1$$.

Understanding Factors of Polynomials

In the previous article, we discussed how to determine which of the given options is a factor of the polynomial $$6x^3 + 6$$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on factors of polynomials.

Q: What is a factor of a polynomial?

A: A factor of a polynomial is an expression that can be multiplied by another expression to produce the original polynomial. In other words, if we have a polynomial $$p(x)$$ and an expression $$f(x)$$, then $$f(x)$$ is a factor of $$p(x)$$ if there exists an expression $$g(x)$$ such that $$p(x) = f(x) \cdot g(x)$$.

Q: How do I determine if a polynomial is a factor of another polynomial?

A: To determine if a polynomial is a factor of another polynomial, we can perform polynomial division. If the remainder of the polynomial division is zero, then the polynomial is a factor of the other polynomial.

Q: What is polynomial division?

A: Polynomial division is a process of dividing one polynomial by another polynomial to obtain a quotient and a remainder. The quotient is the result of the division, and the remainder is the amount left over after the division.

Q: How do I perform polynomial division?

A: To perform polynomial division, we can use the following steps:

Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from step 1.
Subtract the result from step 2 from the dividend.
Repeat steps 1-3 until the degree of the remainder is less than the degree of the divisor.

Q: What is the difference between a factor and a multiple?

A: A factor is an expression that can be multiplied by another expression to produce the original polynomial, while a multiple is an expression that can be obtained by multiplying the original polynomial by another expression.

Q: Can a polynomial have multiple factors?

A: Yes, a polynomial can have multiple factors. For example, the polynomial $$x^2 + 4x + 4$$ can be factored as $$(x + 2)^2$$, which has two factors: $$x + 2$$ and $$x + 2$$.

Q: Can a polynomial have no factors?

A: Yes, a polynomial can have no factors. For example, the polynomial $$x^2 + 1$$ has no real factors, as it cannot be expressed as the product of two real polynomials.

Q: How do I find the factors of a polynomial?

A: To find the factors of a polynomial, we can use various methods, such as:

Factoring by grouping: This involves grouping the terms of the polynomial into pairs and factoring each pair.
Factoring by difference of squares: This involves factoring the polynomial as the difference of two squares.
Factoring by sum and difference: This involves factoring the polynomial as the sum or difference of two expressions.

Q: Can I use technology to find the factors of a polynomial?

A: Yes, you can use technology, such as calculators or computer software, to find the factors of a polynomial. Many calculators and computer software programs have built-in functions for factoring polynomials.

Conclusion

In this Q&A article, we have provided answers to common questions about factors of polynomials. We have discussed how to determine if a polynomial is a factor of another polynomial, how to perform polynomial division, and how to find the factors of a polynomial. We have also provided information on the difference between a factor and a multiple, and how to use technology to find the factors of a polynomial.