Understanding Radical Equations and Polynomial Solutions

The video tutorial explains how to solve an equation involving square and cube roots by converting them into exponential fractions. It demonstrates the process of raising both sides to a common power to eliminate fractional exponents, expanding expressions, and simplifying the equation. The tutorial uses synthetic division and the quadratic formula to find solutions and verifies them to ensure correctness.

What is the first step in solving the equation involving square and cube roots?

• Multiply the radicals

• Convert the radicals into exponential fractions

• Add the radicals together

• Subtract the radicals

What is the least common multiple of the indices 2 and 3 used to eliminate fractional exponents?

• 4

• 5

• 6

• 7

Which formula is used to expand the expression (x + 4)^3?

• a^3 + 3a^2b + 3ab^2 + b^3

• a^2 + 2ab + b^2

• a^2 - 2ab + b^2

• a^3 - 3a^2b + 3ab^2 - b^3

What is the result of expanding (4x + 7)^2?

• 12x^2 + 48x + 36

• 8x^2 + 28x + 14

• 16x^2 + 56x + 49

• 4x^2 + 14x + 7

What is the simplified form of the cubic polynomial derived from the equation?

• x^3 + 3x^2 + 7x - 12

• x^3 + 4x^2 + 8x - 15

• x^3 - 3x^2 - 7x + 12

• x^3 - 4x^2 - 8x + 15

Which method is used to find the roots of the cubic polynomial?

• Graphical method

• Long division

• Synthetic division

• Substitution method

What is the first solution found using synthetic division?

• x = 5

• x = 1

• x = 3

• x = 7

What is the quadratic equation obtained after factoring the cubic polynomial?

• x^2 + x - 3

• x^2 - 2x + 5

• x^2 - x + 3

• x^2 + 2x - 5

Which formula is used to find the remaining solutions of the quadratic equation?

• Graphical method

• Completing the square

• Factorization

• Quadratic formula

What are the verified solutions to the original equation?

• x = 5 and x = 1.3028

• x = 5 and x = -1.3028

• x = 3 and x = 1.3028

• x = 3 and x = -1.3028