Questions and Answers
What is the purpose of using the Factor Theorem in solving cubic equations?
What is the general form of a cubic equation?
What is the expression for the solutions of a quadratic equation using the Quadratic Formula?
What is the purpose of dividing the cubic polynomial by a factor found using the Factor Theorem?
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What is the expression for a cubic polynomial after factorization?
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What is the ultimate goal of solving a cubic equation?
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Study Notes
Solving Cubic Equations
Key Concepts
- Cubic equations have the form
ax^3 + bx^2 + cx + d = 0
- Factorization methods are used to solve cubic equations
- Factor Theorem and Quadratic Formula are essential tools in solving cubic equations
Factor Theorem
- If
f(d/c) = 0
, thencx - d
is a factor ofp(x)
- Helps identify a factor of the cubic polynomial
Quadratic Formula
- Used to solve quadratic expressions
ax^2 + bx + c = 0
- Formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Steps to Solve Cubic Equations
Identify a Factor
- Use the Factor Theorem to find a factor by trial and error
- Substitute potential roots into the polynomial to check if
f(x) = 0
- Potential roots are based on the factors of the constant term
Factorize the Polynomial
- Divide the cubic polynomial by the identified factor
- The cubic polynomial
f(x)
can be expressed as(cx - d) · Q(x)
-
Q(x)
is the quadratic polynomial obtained after division
Solve the Quadratic Polynomial
- Use the quadratic formula to solve the quadratic polynomial
Q(x) = 0
Combine Solutions
- The solutions of the cubic equation are the roots obtained from factorization and solving the quadratic polynomial
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Description
Learn how to solve cubic equations using factorization methods, Factor Theorem, and Quadratic Formula. Understand the concepts and formulas to solve quadratic expressions.