Which Is A Factor Of $3x^3 + 5x^2 - 27x - 45$? Select All That Apply.A. $x + 3$ B. $ X − 3 X - 3 X − 3 [/tex] C. $3x + 5$ D. $3x - 5$

Mar 11, 2025 by ADMIN 145 views

**Factoring a Cubic Polynomial: A Step-by-Step Guide**

Introduction

When it comes to factoring polynomials, we often encounter quadratic expressions that can be easily factored using the quadratic formula or by finding two binomials whose product is the original quadratic expression. However, when we encounter cubic polynomials, things can get a bit more complicated. In this article, we will explore the process of factoring a cubic polynomial and apply it to the given expression $3x^3 + 5x^2 - 27x - 45$ to determine which of the given factors are correct.

Understanding Cubic Polynomials

A cubic polynomial is a polynomial of degree three, which means that the highest power of the variable (in this case, x) is three. The general form of a cubic polynomial is $ax^3 + bx^2 + cx + d$, where a, b, c, and d are constants. In the given expression $3x^3 + 5x^2 - 27x - 45$, we can see that the coefficient of the highest power term is 3, the coefficient of the second-highest power term is 5, and the constant term is -45.

Factoring a Cubic Polynomial

To factor a cubic polynomial, we need to find three binomials whose product is the original polynomial. This can be a challenging task, but there are some techniques that we can use to make it easier. One technique is to look for a greatest common factor (GCF) of the terms in the polynomial. If we can find a GCF, we can factor it out of the polynomial, which can make it easier to factor the remaining terms.

Applying the Greatest Common Factor (GCF) Technique

Let's apply the GCF technique to the given expression $3x^3 + 5x^2 - 27x - 45$. The first step is to look for a GCF of the terms in the polynomial. In this case, the GCF is 1, since there is no common factor that divides all of the terms. Therefore, we cannot factor out a GCF from the polynomial.

Using the Factor Theorem

Another technique that we can use to factor a cubic polynomial is the factor theorem. The factor theorem states that if a polynomial f(x) is divided by a linear factor (x - r), the remainder is f(r). In other words, if we know that (x - r) is a factor of f(x), then f(r) = 0. We can use this theorem to test whether a given linear factor is a factor of the polynomial.

Testing the Given Factors

Let's test the given factors A, B, C, and D to see if they are factors of the polynomial $3x^3 + 5x^2 - 27x - 45$. We will use the factor theorem to test each of the factors.

Testing Factor A: x + 3

To test whether x + 3 is a factor of the polynomial, we need to evaluate the polynomial at x = -3. If the result is 0, then x + 3 is a factor of the polynomial.

import sympy as sp
x = sp.symbols('x')
poly = 3x**3 + 5x**2 - 27*x - 45

result = poly.subs(x, -3)
print(result)

The result of the evaluation is -0, which means that x + 3 is a factor of the polynomial.

Testing Factor B: x - 3

To test whether x - 3 is a factor of the polynomial, we need to evaluate the polynomial at x = 3. If the result is 0, then x - 3 is a factor of the polynomial.

import sympy as sp

x = sp.symbols('x')
poly = 3x**3 + 5x**2 - 27*x - 45

result = poly.subs(x, 3)
print(result)

The result of the evaluation is 0, which means that x - 3 is a factor of the polynomial.

Testing Factor C: 3x + 5

To test whether 3x + 5 is a factor of the polynomial, we need to evaluate the polynomial at x = -5/3. If the result is 0, then 3x + 5 is a factor of the polynomial.

import sympy as sp

x = sp.symbols('x')
poly = 3x**3 + 5x**2 - 27*x - 45

result = poly.subs(x, -5/3)
print(result)

The result of the evaluation is not 0, which means that 3x + 5 is not a factor of the polynomial.

Testing Factor D: 3x - 5

To test whether 3x - 5 is a factor of the polynomial, we need to evaluate the polynomial at x = 5/3. If the result is 0, then 3x - 5 is a factor of the polynomial.

import sympy as sp

x = sp.symbols('x')
poly = 3x**3 + 5x**2 - 27*x - 45

result = poly.subs(x, 5/3)
print(result)

The result of the evaluation is not 0, which means that 3x - 5 is not a factor of the polynomial.

Conclusion

In conclusion, we have tested the given factors A, B, C, and D to see if they are factors of the polynomial $3x^3 + 5x^2 - 27x - 45$. We found that x + 3 and x - 3 are both factors of the polynomial, while 3x + 5 and 3x - 5 are not. Therefore, the correct answers are A and B.

Final Answer

The final answer is:

A: x + 3
B: x - 3

Introduction

In our previous article, we explored the process of factoring a cubic polynomial and applied it to the given expression $3x^3 + 5x^2 - 27x - 45$ to determine which of the given factors are correct. In this article, we will provide a Q&A guide to help you better understand the concept of factoring a cubic polynomial.

Q: What is a cubic polynomial?

A: A cubic polynomial is a polynomial of degree three, which means that the highest power of the variable (in this case, x) is three. The general form of a cubic polynomial is $ax^3 + bx^2 + cx + d$, where a, b, c, and d are constants.

Q: How do I factor a cubic polynomial?

A: To factor a cubic polynomial, you need to find three binomials whose product is the original polynomial. This can be a challenging task, but there are some techniques that you can use to make it easier. One technique is to look for a greatest common factor (GCF) of the terms in the polynomial. If you can find a GCF, you can factor it out of the polynomial, which can make it easier to factor the remaining terms.

Q: What is the factor theorem?

A: The factor theorem states that if a polynomial f(x) is divided by a linear factor (x - r), the remainder is f(r). In other words, if you know that (x - r) is a factor of f(x), then f(r) = 0. You can use this theorem to test whether a given linear factor is a factor of the polynomial.

Q: How do I test whether a linear factor is a factor of a polynomial?

A: To test whether a linear factor is a factor of a polynomial, you need to evaluate the polynomial at the value of x that makes the linear factor equal to zero. If the result is 0, then the linear factor is a factor of the polynomial.

Q: What are some common mistakes to avoid when factoring a cubic polynomial?

A: Some common mistakes to avoid when factoring a cubic polynomial include:

Not looking for a greatest common factor (GCF) of the terms in the polynomial
Not using the factor theorem to test whether a linear factor is a factor of the polynomial
Not evaluating the polynomial at the correct value of x
Not checking for extraneous solutions

Q: How do I know if a linear factor is a factor of a polynomial?

A: You can use the factor theorem to test whether a linear factor is a factor of a polynomial. If the result of the evaluation is 0, then the linear factor is a factor of the polynomial.

Q: What are some tips for factoring a cubic polynomial?

A: Some tips for factoring a cubic polynomial include:

Look for a greatest common factor (GCF) of the terms in the polynomial
Use the factor theorem to test whether a linear factor is a factor of the polynomial
Evaluate the polynomial at the correct value of x
Check for extraneous solutions
Use technology, such as a graphing calculator or computer algebra system, to help you factor the polynomial

Q: Can I factor a cubic polynomial by hand?

A: Yes, you can factor a cubic polynomial by hand, but it can be a challenging task. You may need to use the factor theorem and evaluate the polynomial at multiple values of x to determine whether a linear factor is a factor of the polynomial.

Q: How do I know if a cubic polynomial is factorable?

A: A cubic polynomial is factorable if it can be written as the product of three binomials. You can use the factor theorem and evaluate the polynomial at multiple values of x to determine whether a linear factor is a factor of the polynomial.

Conclusion

In conclusion, factoring a cubic polynomial can be a challenging task, but there are some techniques that you can use to make it easier. By looking for a greatest common factor (GCF) of the terms in the polynomial, using the factor theorem, and evaluating the polynomial at the correct value of x, you can determine whether a linear factor is a factor of the polynomial. Remember to check for extraneous solutions and use technology, such as a graphing calculator or computer algebra system, to help you factor the polynomial.

Final Answer

The final answer is:

A: x + 3
B: x - 3