Factoring the Sum of Cubes in Algebra II

The video tutorial explains how to factor the expression 27x^6 + 125 by recognizing it as a sum of cubes. The instructor introduces the special form for factoring sums of cubes and demonstrates the algebraic multiplication process. By identifying the cube roots of the terms, the expression is rewritten and factored using the special form. The tutorial concludes by emphasizing the importance of recognizing this pattern for successful factoring.

What is the first step in recognizing how to factor the given expression?

• Converting it into a difference of squares

• Identifying it as a sum of cubes

• Applying the quadratic formula

• Performing long division

What does the special form a^3 + b^3 factor into?

• a + b times a^2 + ab + b^2

• a - b times a^2 - ab + b^2

• a - b times a^2 + ab + b^2

• a + b times a^2 - ab + b^2

What property is used to factor a^3 + b^3?

• Factor by grouping

• Difference of squares

• Sum of cubes

• Quadratic formula

What is the cube root of 27x^6?

• 3x^4

• 9x^2

• 3x^2

• 9x^4

What is the result of (3x^2)^2?

• 9x^2

• 6x^2

• 9x^4

• 6x^4

How do you determine the cube root of a term?

• Subtract 3 from the exponent

• Add 3 to the exponent

• Multiply the exponent by 3

• Divide the exponent by 3

What does the expression 3x^2 + 5 represent in the factoring process?

• The difference of cubes

• The sum of the cube roots

• The product of a and b

• The quadratic formula

What is the final factored form of 27x^6 + 125?

• 3x^2 + 5 times 9x^4 - 15x^2 + 25

• 3x^2 - 5 times 9x^4 + 15x^2 - 25

• 3x^2 + 5 times 9x^4 + 15x^2 + 25

• 3x^2 - 5 times 9x^4 - 15x^2 + 25

What is the significance of the expression 9x^4 - 15x^2 + 25?

• It is the integral of the original expression

• It is the derivative of the original expression

• It represents the special form a^2 - ab + b^2

• It is the expanded form of the original expression

Why can't the final factored form of 27x^6 + 125 be factored further using real numbers?

• It is already in its simplest form

• It is a prime number

• It does not contain any variables

• It is an equation, not an expression