If $f(x) = 2x^3 + 10x^2 + 14x - 26$ And $f(1) = 0$, Then Find The Two Remaining Zeros.

Introduction

In algebra, a cubic polynomial is a polynomial of degree three, which means the highest power of the variable is three. Cubic polynomials can be written in the form $ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants. In this article, we will discuss how to find the remaining zeros of a cubic polynomial given one of its zeros.

The Given Polynomial

The given polynomial is $f(x) = 2x^3 + 10x^2 + 14x - 26$. We are also given that $f(1) = 0$, which means that $x = 1$ is a zero of the polynomial.

The Factor Theorem

The factor theorem states that if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$. In this case, since $f(1) = 0$, we know that $(x - 1)$ is a factor of $f(x)$.

Dividing the Polynomial

To find the remaining zeros of the polynomial, we need to divide the polynomial by $(x - 1)$. We can do this using long division or synthetic division.

import sympy as sp

# Define the polynomial
x = sp.symbols('x')
f = 2*x**3 + 10*x**2 + 14*x - 26

# Divide the polynomial by (x - 1)
g = sp.div(f, x - 1)

print(g)

This will give us the quotient $g(x) = 2x^2 + 12x + 26$.

Finding the Remaining Zeros

To find the remaining zeros of the polynomial, we need to solve the equation $g(x) = 0$. We can do this by factoring the quadratic expression or using the quadratic formula.

import sympy as sp

# Define the quotient
x = sp.symbols('x')
g = 2*x**2 + 12*x + 26

# Solve the equation g(x) = 0
solutions = sp.solve(g, x)

print(solutions)

This will give us the remaining zeros of the polynomial.

Conclusion

In this article, we discussed how to find the remaining zeros of a cubic polynomial given one of its zeros. We used the factor theorem and polynomial division to find the quotient, and then solved the equation to find the remaining zeros. The two remaining zeros of the polynomial are $\boxed{-3}$ and $\boxed{-\frac{13}{2}}$.

The Final Answer

The final answer is $\boxed{-3, -\frac{13}{2}}$.

Additional Information

• The factor theorem can be used to find the zeros of a polynomial given one of its factors.
• Polynomial division can be used to find the quotient of a polynomial divided by another polynomial.
• The quadratic formula can be used to solve quadratic equations.

References

• Sympy Documentation
• Wikipedia: Factor Theorem
• Wikipedia: Polynomial Division
• Wikipedia: Quadratic Formula
  Q&A: Finding the Remaining Zeros of a Cubic Polynomial =====================================================

Introduction

In our previous article, we discussed how to find the remaining zeros of a cubic polynomial given one of its zeros. In this article, we will answer some frequently asked questions related to finding the remaining zeros of a cubic polynomial.

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 know one of the zeros of a polynomial, we can write it as a factor of the polynomial.

Q: How do I use the factor theorem to find the remaining zeros of a cubic polynomial?

A: To use the factor theorem, we need to divide the polynomial by the known factor. We can do this using long division or synthetic division. Once we have the quotient, we can solve the equation to find the remaining zeros.

Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial by another polynomial. It is used to find the quotient of a polynomial divided by another polynomial.

Q: How do I perform polynomial division?

A: There are two main methods of performing polynomial division: long division and synthetic division. Long division is a more traditional method, while synthetic division is a faster and more efficient method.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to find the remaining zeros of a cubic polynomial?

A: To use the quadratic formula, we need to rewrite the quadratic expression in the form $ax^2 + bx + c$. We can then plug the values of $a$, $b$, and $c$ into the quadratic formula to find the remaining zeros.

Q: What are some common mistakes to avoid when finding the remaining zeros of a cubic polynomial?

A: Some common mistakes to avoid when finding the remaining zeros of a cubic polynomial include:

• Not using the correct method of polynomial division
• Not checking the quotient for any remaining factors
• Not solving the equation correctly
• Not checking the solutions for any extraneous solutions

Q: How do I check my solutions for any extraneous solutions?

A: To check your solutions for any extraneous solutions, you can plug the solutions back into the original equation to see if they are true. If they are not true, then they are extraneous solutions.

Q: What are some real-world applications of finding the remaining zeros of a cubic polynomial?

A: Some real-world applications of finding the remaining zeros of a cubic polynomial include:

• Modeling population growth
• Modeling the motion of objects
• Finding the maximum or minimum value of a function

Conclusion

In this article, we answered some frequently asked questions related to finding the remaining zeros of a cubic polynomial. We discussed the factor theorem, polynomial division, and the quadratic formula, and provided some tips and tricks for avoiding common mistakes. We also discussed some real-world applications of finding the remaining zeros of a cubic polynomial.

Additional Information

• The factor theorem can be used to find the zeros of a polynomial given one of its factors.
• Polynomial division can be used to find the quotient of a polynomial divided by another polynomial.
• The quadratic formula can be used to solve quadratic equations.

References

• Sympy Documentation
• Wikipedia: Factor Theorem
• Wikipedia: Polynomial Division
• Wikipedia: Quadratic Formula