The video tutorial explains how to solve a polynomial division problem using three different methods. The problem involves dividing 8x^3 - 27y^3 by 2x - 3y and equating it to ax^2 + bxy + cy^2. The first method involves direct division, the second method uses cross multiplication, and the third method applies cube identities. Each method leads to the same result, determining the values of a, b, and c, and ultimately calculating 2a + b - c, which equals 5.

What is the initial expression given in the problem?

• 8x^3 - 27y^3 = ax^2 + bx + cy^2

• 8x^3 - 27y^3 = ax^2 + bxy + cy^2

• 8x^3 - 27y^3 = ax^2 + bxy + cy

• 8x^3 - 27y^3 = ax + bxy + cy^2

In the first method, what is the result of dividing 8x^3 by 2x?

• 4x^2

• 3x^2

• 6xy

• 9y^2

What is the value of 'a' found using the first method?

• 5

• 4

• 3

• 6

What is the final value of 2a + b - c using the first method?

• 6

• 5

• 4

• 3

In the second method, what operation is performed after setting up the equation?

• Polynomial division

• Cross multiplication

• Integration

• Substitution

What is the value of 'c' found using the second method?

• 7

• 8

• 9

• 10

What is the relationship between 2b and 3a in the second method?

• 2b = 5a

• 2b = 6a

• 2b = 4a

• 2b = 3a

What is the final value of 2a + b - c using the second method?

• 3

• 6

• 4

• 5

In the third method, which identity is used to simplify the expression?

• Sum of cubes

• Sum of squares

• Difference of squares

• Difference of cubes

What is the expression for a cube minus b cube?

• (a - b)(a^2 + b^2 + ab)

• (a + b)(a^2 + b^2 - ab)

• (a + b)(a^2 - b^2 + ab)

• (a - b)(a^2 - b^2 - ab)