Podcast
Questions and Answers
Find all the roots of the equation $X^3 + x^2 + 4x + 4 = 0$.
Find all the roots of the equation $X^3 + x^2 + 4x + 4 = 0$.
-1, +2i
Find all the roots of the equation $X^3 + 4x^2 + x - 6 = 0$.
Find all the roots of the equation $X^3 + 4x^2 + x - 6 = 0$.
-3, -2, 1
What are the roots of the function?
What are the roots of the function?
2, +√3
What are the roots of the function?
What are the roots of the function?
Signup and view all the answers
What are the roots of the function?
What are the roots of the function?
Signup and view all the answers
What are the roots of the function?
What are the roots of the function?
Signup and view all the answers
What are the roots of the function?
What are the roots of the function?
Signup and view all the answers
Find all the roots of the equation $X^3+x^2+4x+4=0$.
Find all the roots of the equation $X^3+x^2+4x+4=0$.
Signup and view all the answers
Find the roots of the equation $X^3+4x^2+x-6=0$.
Find the roots of the equation $X^3+4x^2+x-6=0$.
Signup and view all the answers
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Signup and view all the answers
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Signup and view all the answers
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Signup and view all the answers
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Signup and view all the answers
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Find the roots of a polynomial that satisfies the conditions provided in the definition.
Signup and view all the answers
Determine whether the following statement is always, sometimes, or never true: A polynomial function with real coefficients has real zeros.
Determine whether the following statement is always, sometimes, or never true: A polynomial function with real coefficients has real zeros.
Signup and view all the answers
Determine whether the following statement is always, sometimes, or never true: A polynomial function that does not intercept the X-axis has complex roots only.
Determine whether the following statement is always, sometimes, or never true: A polynomial function that does not intercept the X-axis has complex roots only.
Signup and view all the answers
Determine whether the following statement is always, sometimes, or never true: An odd degree polynomial function with real coefficients has at least one real root.
Determine whether the following statement is always, sometimes, or never true: An odd degree polynomial function with real coefficients has at least one real root.
Signup and view all the answers
Study Notes
Roots of Polynomial Equations
- The equation ( X^3 + X^2 + 4X + 4 = 0 ) has roots at -1 and ( +2i ).
- The equation ( X^3 + 4X^2 + X - 6 = 0 ) has roots at -3, -2, and 1.
- A third example yields roots at 2 and ( +\sqrt{3} ).
- An additional equation has roots at ( +2 ) and ( +\sqrt{2} ).
- The equation with roots at -6 and ( +i ).
- Another polynomial has roots at -2 and ( 3 + i\sqrt{2} ).
- Another polynomial found roots at -2, 3, and ( +i ).
Statements About Polynomial Functions
- A polynomial function with real coefficients has real zeros sometimes.
- A polynomial function that does not intercept the x-axis always has complex roots only.
- All polynomial functions of degree greater than zero with real coefficients always have at least one real root.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of the Fundamental Theorem of Algebra with these practice flashcards. Solve various polynomial equations and find their roots without using a calculator. This quiz will help reinforce your skills in identifying complex and real roots.
