Select All Of The Following That Are Potential Roots Of P ( X ) = X 4 − 9 X 2 − 4 X + 12 P(x) = X^4 - 9x^2 - 4x + 12 P ( X ) = X 4 − 9 X 2 − 4 X + 12 .A. 0 B. ± 2 \pm 2 ± 2 C. ± 4 \pm 4 ± 4 D. ± 9 \pm 9 ± 9 E. 1 2 \frac{1}{2} 2 1 ​ F. ± 3 \pm 3 ± 3 G. ± 6 \pm 6 ± 6 H. ± 12 \pm 12 ± 12

Introduction

In algebra, finding the roots of a polynomial is a crucial step in solving equations and understanding the behavior of functions. A root of a polynomial is a value of x that makes the polynomial equal to zero. In this article, we will explore how to find the roots of a given polynomial, specifically the polynomial $p(x) = x^4 - 9x^2 - 4x + 12$. We will examine the potential roots of this polynomial and determine which ones are actually roots.

The Rational Root Theorem

The Rational Root Theorem states that if a rational number p/q is a root of the polynomial $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, then p must be a factor of the constant term $a_0$, and q must be a factor of the leading coefficient $a_n$. In our case, the constant term is 12, and the leading coefficient is 1. Therefore, the potential rational roots of the polynomial are the factors of 12, which are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

The Potential Roots

Based on the Rational Root Theorem, the potential roots of the polynomial $p(x) = x^4 - 9x^2 - 4x + 12$ are:

• $\pm 1$
• $\pm 2$
• $\pm 3$
• $\pm 4$
• $\pm 6$
• $\pm 12$

Testing the Potential Roots

To determine which of these potential roots are actually roots of the polynomial, we need to substitute each value into the polynomial and check if the result is equal to zero.

Substituting x = 0

Let's start by substituting x = 0 into the polynomial:

$p(0) = 0^4 - 9(0)^2 - 4(0) + 12$

$p(0) = 12 \neq 0$

Therefore, x = 0 is not a root of the polynomial.

Substituting x = 1

Next, let's substitute x = 1 into the polynomial:

$p(1) = 1^4 - 9(1)^2 - 4(1) + 12$

$p(1) = 1 - 9 - 4 + 12$

$p(1) = 0$

Therefore, x = 1 is a root of the polynomial.

Substituting x = -1

Now, let's substitute x = -1 into the polynomial:

$p(-1) = (-1)^4 - 9(-1)^2 - 4(-1) + 12$

$p(-1) = 1 - 9 + 4 + 12$

$p(-1) = 8 \neq 0$

Therefore, x = -1 is not a root of the polynomial.

Substituting x = 2

Next, let's substitute x = 2 into the polynomial:

$p(2) = 2^4 - 9(2)^2 - 4(2) + 12$

$p(2) = 16 - 36 - 8 + 12$

$p(2) = -16 \neq 0$

Therefore, x = 2 is not a root of the polynomial.

Substituting x = -2

Now, let's substitute x = -2 into the polynomial:

$p(-2) = (-2)^4 - 9(-2)^2 - 4(-2) + 12$

$p(-2) = 16 - 36 + 8 + 12$

$p(-2) = 0$

Therefore, x = -2 is a root of the polynomial.

Substituting x = 3

Next, let's substitute x = 3 into the polynomial:

$p(3) = 3^4 - 9(3)^2 - 4(3) + 12$

$p(3) = 81 - 81 - 12 + 12$

$p(3) = 0$

Therefore, x = 3 is a root of the polynomial.

Substituting x = -3

Now, let's substitute x = -3 into the polynomial:

$p(-3) = (-3)^4 - 9(-3)^2 - 4(-3) + 12$

$p(-3) = 81 - 81 + 12 + 12$

$p(-3) = 24 \neq 0$

Therefore, x = -3 is not a root of the polynomial.

Substituting x = 4

Next, let's substitute x = 4 into the polynomial:

$p(4) = 4^4 - 9(4)^2 - 4(4) + 12$

$p(4) = 256 - 144 - 16 + 12$

$p(4) = 108 \neq 0$

Therefore, x = 4 is not a root of the polynomial.

Substituting x = -4

Now, let's substitute x = -4 into the polynomial:

$p(-4) = (-4)^4 - 9(-4)^2 - 4(-4) + 12$

$p(-4) = 256 - 144 + 16 + 12$

$p(-4) = 140 \neq 0$

Therefore, x = -4 is not a root of the polynomial.

Substituting x = 6

Next, let's substitute x = 6 into the polynomial:

$p(6) = 6^4 - 9(6)^2 - 4(6) + 12$

$p(6) = 1296 - 324 - 24 + 12$

$p(6) = 960 \neq 0$

Therefore, x = 6 is not a root of the polynomial.

Substituting x = -6

Now, let's substitute x = -6 into the polynomial:

$p(-6) = (-6)^4 - 9(-6)^2 - 4(-6) + 12$

$p(-6) = 1296 - 324 + 24 + 12$

$p(-6) = 1008 \neq 0$

Therefore, x = -6 is not a root of the polynomial.

Substituting x = 12

Next, let's substitute x = 12 into the polynomial:

$p(12) = 12^4 - 9(12)^2 - 4(12) + 12$

$p(12) = 20736 - 1458 - 48 + 12$

$p(12) = 19242 \neq 0$

Therefore, x = 12 is not a root of the polynomial.

Substituting x = -12

Now, let's substitute x = -12 into the polynomial:

$p(-12) = (-12)^4 - 9(-12)^2 - 4(-12) + 12$

$p(-12) = 20736 - 1458 + 48 + 12$

$p(-12) = 19238 \neq 0$

Therefore, x = -12 is not a root of the polynomial.

Conclusion

In conclusion, the potential roots of the polynomial $p(x) = x^4 - 9x^2 - 4x + 12$ are:

• $\pm 1$
• $\pm 2$
• $\pm 3$
• $\pm 4$
• $\pm 6$
• $\pm 12$

After testing each of these potential roots, we found that the actual roots of the polynomial are:

• $x = 1$
• $x = -2$
• $x = 3$

Therefore, the correct answers are:

• B. $\pm 2$
• C. $\pm 4$
• F. $\pm 3$
  

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem in algebra that states that if a rational number p/q is a root of the polynomial $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, then p must be a factor of the constant term $a_0$, and q must be a factor of the leading coefficient $a_n$.

Q: How do I find the potential roots of a polynomial using the Rational Root Theorem?

A: To find the potential roots of a polynomial using the Rational Root Theorem, you need to list all the factors of the constant term and the leading coefficient. Then, you need to take the ratio of each factor of the constant term to each factor of the leading coefficient to get the potential rational roots.

Q: What is the difference between a root and a factor of a polynomial?

A: A root of a polynomial is a value of x that makes the polynomial equal to zero. A factor of a polynomial is a value of x that divides the polynomial evenly, resulting in a remainder of zero.

Q: How do I test if a potential root is actually a root of the polynomial?

A: To test if a potential root is actually a root of the polynomial, you need to substitute the potential root into the polynomial and check if the result is equal to zero.

Q: What if I get a remainder when I substitute the potential root into the polynomial?

A: If you get a remainder when you substitute the potential root into the polynomial, then the potential root is not a root of the polynomial.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with irrational coefficients?

A: No, the Rational Root Theorem only works for polynomials with rational coefficients. If you have a polynomial with irrational coefficients, you will need to use other methods to find the roots.

Q: How do I find the roots of a polynomial that is not a rational root?

A: If the polynomial is not a rational root, you will need to use other methods such as factoring, synthetic division, or numerical methods to find the roots.

Q: Can I use a calculator to find the roots of a polynomial?

A: Yes, you can use a calculator to find the roots of a polynomial. Many calculators have built-in functions for finding roots, such as the "solve" function.

Q: How do I know if a polynomial has real roots or complex roots?

A: To determine if a polynomial has real roots or complex roots, you need to examine the polynomial and look for any complex numbers in the coefficients. If the coefficients are all real numbers, then the polynomial will have real roots. If the coefficients contain complex numbers, then the polynomial will have complex roots.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with complex coefficients?

A: No, the Rational Root Theorem only works for polynomials with rational coefficients. If you have a polynomial with complex coefficients, you will need to use other methods to find the roots.

Q: How do I find the roots of a polynomial with multiple variables?

A: To find the roots of a polynomial with multiple variables, you need to use methods such as substitution, elimination, or numerical methods.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with multiple variables?

A: No, the Rational Root Theorem only works for polynomials with one variable. If you have a polynomial with multiple variables, you will need to use other methods to find the roots.

Q: How do I know if a polynomial has a single root or multiple roots?

A: To determine if a polynomial has a single root or multiple roots, you need to examine the polynomial and look for any repeated factors. If the polynomial has a repeated factor, then it will have multiple roots. If the polynomial does not have any repeated factors, then it will have a single root.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with multiple roots?

A: Yes, you can use the Rational Root Theorem to find the roots of a polynomial with multiple roots. However, you will need to use the theorem multiple times to find all the roots.

Q: How do I find the roots of a polynomial with a degree greater than 4?

A: To find the roots of a polynomial with a degree greater than 4, you need to use methods such as synthetic division, numerical methods, or computer algebra systems.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with a degree greater than 4?

A: No, the Rational Root Theorem only works for polynomials with a degree of 4 or less. If you have a polynomial with a degree greater than 4, you will need to use other methods to find the roots.

Q: How do I know if a polynomial has a rational root or an irrational root?

A: To determine if a polynomial has a rational root or an irrational root, you need to examine the polynomial and look for any rational numbers in the coefficients. If the coefficients contain rational numbers, then the polynomial will have rational roots. If the coefficients do not contain rational numbers, then the polynomial will have irrational roots.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with irrational roots?

A: No, the Rational Root Theorem only works for polynomials with rational roots. If you have a polynomial with irrational roots, you will need to use other methods to find the roots.

Q: How do I find the roots of a polynomial with a complex coefficient?

A: To find the roots of a polynomial with a complex coefficient, you need to use methods such as synthetic division, numerical methods, or computer algebra systems.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with a complex coefficient?

A: No, the Rational Root Theorem only works for polynomials with rational coefficients. If you have a polynomial with a complex coefficient, you will need to use other methods to find the roots.

Q: How do I know if a polynomial has a single root or multiple roots?

A: To determine if a polynomial has a single root or multiple roots, you need to examine the polynomial and look for any repeated factors. If the polynomial has a repeated factor, then it will have multiple roots. If the polynomial does not have any repeated factors, then it will have a single root.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with multiple roots?

A: Yes, you can use the Rational Root Theorem to find the roots of a polynomial with multiple roots. However, you will need to use the theorem multiple times to find all the roots.

Q: How do I find the roots of a polynomial with a degree greater than 4?

A: To find the roots of a polynomial with a degree greater than 4, you need to use methods such as synthetic division, numerical methods, or computer algebra systems.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with a degree greater than 4?

A: No, the Rational Root Theorem only works for polynomials with a degree of 4 or less. If you have a polynomial with a degree greater than 4, you will need to use other methods to find the roots.

Q: How do I know if a polynomial has a rational root or an irrational root?

A: To determine if a polynomial has a rational root or an irrational root, you need to examine the polynomial and look for any rational numbers in the coefficients. If the coefficients contain rational numbers, then the polynomial will have rational roots. If the coefficients do not contain rational numbers, then the polynomial will have irrational roots.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with irrational roots?

A: No, the Rational Root Theorem only works for polynomials with rational roots. If you have a polynomial with irrational roots, you will need to use other methods to find the roots.

Q: How do I find the roots of a polynomial with a complex coefficient?

A: To find the roots of a polynomial with a complex coefficient, you need to use methods such as synthetic division, numerical methods, or computer algebra systems.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with a complex coefficient?

A: No, the Rational Root Theorem only works for polynomials with rational coefficients. If you have a polynomial with a complex coefficient, you will need to use other methods to find the roots.

Q: How do I know if a polynomial has a single root or multiple roots?

A: To determine if a polynomial has a single root or multiple roots, you need to examine the polynomial and look for any repeated factors. If the polynomial has a repeated factor, then it will have multiple roots. If the polynomial does not have any repeated factors, then it will have a single root.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial with multiple roots?

A: Yes, you can use the Rational Root Theorem to find the roots of a polynomial with multiple roots. However, you will need to use the theorem multiple times to find all the roots.

Q: How do I find the roots of a polynomial with a degree greater than 4?

A: To find