Mathematics
Grade 11
15 min
Factor sums and differences of cubes
Factor sums and differences of cubes
What you'll learn
- Identify the multiplication equation that represents a given array by writing the equation with 100% accuracy.
- Solve for the missing factor in a division equation when given a related multiplication equation representing an array in at least 3 out of 4 attempts.
- Explain how multiplication and division are related using arrays by providing a clear explanation with at least two supporting details about rows and columns.
- Apply knowledge of arrays to create both a multiplication and a division equation using the same array, and explain how they are related, achieving success in 2 out of 3 trials.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify binomials that are sums or differences of perfect cubes.
Recall and correctly write the formulas for factoring a³ + b³ and a³ - b³.
Apply the formulas to factor binomials involving integers and single variables.
Factor more complex expressions involving coefficients and variables with exponents that are multiples of three.
Recognize that the resulting trinomial factor from the sum or difference of cubes formula is prime over the real numbers.
Solve polynomial equations by first factoring a sum or difference of cubes.
Ever seen a complex polynomial like x³ + 512 and wondered if there's a secret pattern to break it down? 🤔 There is, and it's an elegant one!
This tutorial will teach you two powerful formulas for factoring binomials that...
2
Key Concepts & Vocabulary
TermDefinitionExample
Perfect CubeA number or expression that is the result of multiplying a number or expression by itself three times.27 is a perfect cube because 3 × 3 × 3 = 3³. Also, 64y⁶ is a perfect cube because (4y²)³ = 64y⁶.
Cube RootThe base number or expression that is multiplied by itself three times to get the perfect cube.The cube root of 125 is 5, written as ³√125 = 5.
BinomialA polynomial expression with exactly two terms, separated by a plus or minus sign.x³ - 8
TrinomialA polynomial expression with exactly three terms.x² + 2x + 4
FactoringThe process of breaking down a polynomial into a product of its factors (simpler polynomials).Factoring x² - 9 gives (x - 3)(x + 3).
Prime PolynomialA polynomial with integer coefficients that cannot be factored into polynomials of a low...
3
Core Formulas
Sum of Cubes Formula
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Use this formula when you have two perfect cubes added together. Identify the cube root of the first term as 'a' and the cube root of the second term as 'b', then substitute them into the formula.
Difference of Cubes Formula
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Use this formula when you have two perfect cubes with a subtraction sign between them. Identify 'a' and 'b' in the same way and substitute into this pattern.
5 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Find all solutions, including complex ones, for the equation x³ + 8 = 0.
A.x = -2
B.x = -2, x = 2i, x = -2i
C.x = -2, x = 1 ± i√3
D.x = 2, x = -1 ± i√3
Challenging
Factor the expression (x + 3)³ + 8 completely.
A.(x + 5)(x² + 4x + 7)
B.(x + 1)(x² + 8x + 19)
C.(x + 5)(x² + x + 1)
D.(x + 5)(x² + 6y + 9)
Challenging
If (3x - 4) is one of the factors of a difference of two perfect cubes, what was the original binomial?
A.9x³ - 16
B.27x³ - 64
C.3x³ - 4
D.27x³ + 64
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing FreeMore from Factoring
Mathematics for other grades
Frequently asked questions
What grade level is "Factor sums and differences of cubes"?
Factor sums and differences of cubes is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Factor sums and differences of cubes?
You'll be able to: Identify the multiplication equation that represents a given array by writing the equation with 100% accuracy; Solve for the missing factor in a division equation when given a related multiplication equation representing an….
Is "Factor sums and differences of cubes" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Factor sums and differences of cubes?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.