Products, Factors and Power Expansions Formula Bank

Special Products

1. Square of a binomial:
\[
(x + y)^2 = x^2 + 2xy + y^2
\]

2. Square of a binomial (with subtraction):
\[
(x – y)^2 = x^2 – 2xy + y^2
\]

3. Cube of a binomial:
\[
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\]

4. Cube of a binomial (with subtraction):
\[
(x – y)^3 = x^3 – 3x^2y + 3xy^2 – y^3
\]

5. Fourth power of a binomial:
\[
(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
\]

6. Fourth power of a binomial (with subtraction):
\[
(x – y)^4 = x^4 – 4x^3y + 6x^2y^2 – 4xy^3 + y^4
\]

7. Fifth power of a binomial:
\[
(x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5
\]

8. Fifth power of a binomial (with subtraction):
\[
(x – y)^5 = x^5 – 5x^4y + 10x^3y^2 – 10x^2y^3 + 5xy^4 – y^5
\]

9. Sixth power of a binomial:
\[
(x + y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6
\]

10. Sixth power of a binomial (with subtraction):
\[
(x – y)^6 = x^6 – 6x^5y + 15x^4y^2 – 20x^3y^3 + 15x^2y^4 – 6xy^5 + y^6
\]

 Factors

1. Difference of squares:
\[
x^2 – y^2 = (x – y)(x + y)
\]

2. Difference of cubes:
\[
x^3 – y^3 = (x – y)(x^2 + xy + y^2)
\]

3. Sum of cubes:
\[
x^3 + y^3 = (x + y)(x^2 – xy + y^2)
\]

4. Difference of fourth powers:
\[
x^4 – y^4 = (x – y)(x + y)(x^2 + y^2)
\]

5. Difference of fifth powers:
\[
x^5 – y^5 = (x – y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)
\]

6. Sum of fifth powers:
\[
x^5 + y^5 = (x + y)(x^4 – x^3y + x^2y^2 – xy^3 + y^4)
\]

7. Difference of sixth powers:
\[
x^6 – y^6 = (x – y)(x + y)(x^2 + y^2)(x^2 – xy + y^2)
\]