Solving Cubic Equations with Factor Theorem and Quadratic Formula

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Questions and Answers

What is the purpose of using the Factor Theorem in solving cubic equations?

  • To identify a possible factor of the polynomial (correct)
  • To find the exact solution of the cubic equation
  • To simplify the cubic equation
  • To apply the Quadratic Formula

What is the general form of a cubic equation?

  • \( ax^4 + bx^3 + cx^2 + dx = 0 \)
  • \( ax^2 + bx + c + d = 0 \)
  • \( ax^3 + bx^2 + cx + d = 0 \) (correct)
  • \( ax^2 + bx + c = 0 \)

What is the expression for the solutions of a quadratic equation using the Quadratic Formula?

  • \( x = rac{b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • \( x = rac{-b \mp \sqrt{b^2 + 4ac}}{a} \)
  • \( x = rac{-b \pm \sqrt{b^2 - 4ac}}{a} \) (correct)
  • \( x = rac{-b \pm \sqrt{b^2 + 4ac}}{2a} \)

What is the purpose of dividing the cubic polynomial by a factor found using the Factor Theorem?

<p>To obtain a quadratic polynomial (D)</p>
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What is the expression for a cubic polynomial after factorization?

<p>( f(x) = (cx - d) \cdot Q(x) ) (A)</p>
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What is the ultimate goal of solving a cubic equation?

<p>To find the roots of the equation (D)</p>
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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, then cx - d is a factor of p(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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