Select The Correct Answer.What Are The Zeros Of G ( X ) = X 3 + 6 X 2 − 9 X − 54 G(x) = X^3 + 6x^2 - 9x - 54 G ( X ) = X 3 + 6 X 2 − 9 X − 54 ?A. 1 , 2 , 27 1, 2, 27 1 , 2 , 27 B. 3 , − 3 , − 6 3, -3, -6 3 , − 3 , − 6 C. − 6 , 3 , 6 -6, 3, 6 − 6 , 3 , 6 D. 2 , − 1 , 18 2, -1, 18 2 , − 1 , 18

Introduction

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified based on the degree of the polynomial, which is the highest power of the variable. A cubic polynomial is a polynomial of degree three, meaning it has the highest power of three. In this article, we will focus on finding the zeros of a cubic polynomial, specifically the polynomial $g(x) = x^3 + 6x^2 - 9x - 54$.

What are Zeros in a Polynomial?

In the context of polynomials, a zero is a value of the variable that makes the polynomial equal to zero. In other words, if we substitute a zero into the polynomial, the result will be zero. Finding the zeros of a polynomial is an essential concept in algebra, as it helps us understand the behavior of the polynomial and its graph.

Factoring the Cubic Polynomial

To find the zeros of the cubic polynomial $g(x) = x^3 + 6x^2 - 9x - 54$, we can start by factoring the polynomial. Factoring involves expressing the polynomial as a product of simpler polynomials. In this case, we can try to factor the polynomial by grouping terms.

g(x) = x^3 + 6x^2 - 9x - 54
= (x^3 + 6x^2) - (9x + 54)
= x^2(x + 6) - 9(x + 6)
= (x + 6)(x^2 - 9)

Factoring the Quadratic Polynomial

Now that we have factored the cubic polynomial into a product of a linear polynomial and a quadratic polynomial, we can focus on factoring the quadratic polynomial. The quadratic polynomial $x^2 - 9$ can be factored as a difference of squares.

x^2 - 9 = (x + 3)(x - 3)

Finding the Zeros of the Polynomial

Now that we have factored the cubic polynomial into a product of three linear polynomials, we can find the zeros of the polynomial by setting each factor equal to zero and solving for the variable.

(x + 6)(x + 3)(x - 3) = 0
x + 6 = 0 --> x = -6
x + 3 = 0 --> x = -3
x - 3 = 0 --> x = 3

Conclusion

In conclusion, we have found the zeros of the cubic polynomial $g(x) = x^3 + 6x^2 - 9x - 54$ by factoring the polynomial and setting each factor equal to zero. The zeros of the polynomial are $x = -6, x = -3,$ and $x = 3$. This means that the polynomial has three real zeros, which are the values of the variable that make the polynomial equal to zero.

Discussion and Analysis

The zeros of a polynomial are an essential concept in algebra, as they help us understand the behavior of the polynomial and its graph. In this article, we have focused on finding the zeros of a cubic polynomial, specifically the polynomial $g(x) = x^3 + 6x^2 - 9x - 54$. We have used factoring to find the zeros of the polynomial, and we have found that the zeros are $x = -6, x = -3,$ and $x = 3$.

Real-World Applications

The concept of zeros in polynomials has many real-world applications. For example, in physics, the zeros of a polynomial can represent the frequencies at which a system vibrates. In engineering, the zeros of a polynomial can represent the stability of a system. In economics, the zeros of a polynomial can represent the equilibrium prices of a market.

Conclusion

In conclusion, finding the zeros of a cubic polynomial is an essential concept in algebra. By factoring the polynomial and setting each factor equal to zero, we can find the zeros of the polynomial. The zeros of a polynomial are an essential concept in many real-world applications, including physics, engineering, and economics.

Final Answer

The final answer to the problem is:

• A. $1, 2, 27$: This is incorrect, as the zeros of the polynomial are $x = -6, x = -3,$ and $x = 3$.
• B. $3, -3, -6$: This is correct, as the zeros of the polynomial are $x = -6, x = -3,$ and $x = 3$.
• C. $-6, 3, 6$: This is incorrect, as the zeros of the polynomial are $x = -6, x = -3,$ and $x = 3$.
• D. $2, -1, 18$: This is incorrect, as the zeros of the polynomial are $x = -6, x = -3,$ and $x = 3$.

The correct answer is B. $3, -3, -6$.

Introduction

In our previous article, we discussed how to find the zeros of a cubic polynomial by factoring the polynomial and setting each factor equal to zero. In this article, we will answer some common questions related to finding the zeros of a cubic polynomial.

Q: What is a cubic polynomial?

A: A cubic polynomial is a polynomial of degree three, meaning it has the highest power of three. It is a polynomial that can be written in the form $ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is the variable.

Q: How do I find the zeros of a cubic polynomial?

A: To find the zeros of a cubic polynomial, you can start by factoring the polynomial. If the polynomial cannot be factored, you can use other methods such as the Rational Root Theorem or synthetic division to find the zeros.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem that states that if a rational number $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term of $f(x)$, and $q$ must be a factor of the leading coefficient of $f(x)$.

Q: How do I use synthetic division to find the zeros of a polynomial?

A: Synthetic division is a method of dividing a polynomial by a linear factor. To use synthetic division to find the zeros of a polynomial, you can divide the polynomial by a linear factor that you suspect may be a zero of the polynomial.

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

A: A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. A root of a polynomial is a value of the variable that makes the polynomial equal to zero, but it can also be a complex number.

Q: Can a polynomial have more than one zero?

A: Yes, a polynomial can have more than one zero. In fact, a polynomial of degree $n$ can have up to $n$ zeros.

Q: How do I know if a polynomial has a zero that is not a rational number?

A: If a polynomial has a zero that is not a rational number, it means that the zero is an irrational number or a complex number. You can use methods such as the quadratic formula or synthetic division to find the zeros of the polynomial.

Q: Can a polynomial have a zero that is a complex number?

A: Yes, a polynomial can have a zero that is a complex number. In fact, a polynomial of degree $n$ can have up to $n$ complex zeros.

Q: How do I find the zeros of a polynomial with complex coefficients?

A: To find the zeros of a polynomial with complex coefficients, you can use methods such as the quadratic formula or synthetic division. You can also use the fact that the zeros of a polynomial with complex coefficients are the complex conjugates of each other.

Conclusion

In conclusion, finding the zeros of a cubic polynomial is an essential concept in algebra. By factoring the polynomial and setting each factor equal to zero, we can find the zeros of the polynomial. We have also discussed some common questions related to finding the zeros of a cubic polynomial, including the Rational Root Theorem, synthetic division, and complex zeros.

Final Answer

The final answer to the problem is:

• Q: What is a cubic polynomial?
  • A: A cubic polynomial is a polynomial of degree three, meaning it has the highest power of three.
• Q: How do I find the zeros of a cubic polynomial?
  • A: To find the zeros of a cubic polynomial, you can start by factoring the polynomial. If the polynomial cannot be factored, you can use other methods such as the Rational Root Theorem or synthetic division to find the zeros.
• Q: What is the Rational Root Theorem?
  • A: The Rational Root Theorem is a theorem that states that if a rational number $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term of $f(x)$, and $q$ must be a factor of the leading coefficient of $f(x)$.
• Q: How do I use synthetic division to find the zeros of a polynomial?
  • A: Synthetic division is a method of dividing a polynomial by a linear factor. To use synthetic division to find the zeros of a polynomial, you can divide the polynomial by a linear factor that you suspect may be a zero of the polynomial.
• Q: What is the difference between a zero and a root of a polynomial?
  • A: A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. A root of a polynomial is a value of the variable that makes the polynomial equal to zero, but it can also be a complex number.
• Q: Can a polynomial have more than one zero?
  • A: Yes, a polynomial can have more than one zero. In fact, a polynomial of degree $n$ can have up to $n$ zeros.
• Q: How do I know if a polynomial has a zero that is not a rational number?
  • A: If a polynomial has a zero that is not a rational number, it means that the zero is an irrational number or a complex number. You can use methods such as the quadratic formula or synthetic division to find the zeros of the polynomial.
• Q: Can a polynomial have a zero that is a complex number?
  • A: Yes, a polynomial can have a zero that is a complex number. In fact, a polynomial of degree $n$ can have up to $n$ complex zeros.
• Q: How do I find the zeros of a polynomial with complex coefficients?
  • A: To find the zeros of a polynomial with complex coefficients, you can use methods such as the quadratic formula or synthetic division. You can also use the fact that the zeros of a polynomial with complex coefficients are the complex conjugates of each other.