Algebraic Identities or Notable Products. Among the most important are: a binomial square, a binomial cube, conjugated binomials, binomials with a common term, binomials with a similar term.
Summary
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- 1 Definition
- 2 Square of a binomial
- 3 conjugated binomials
- 4 Binomials with common term
- 5 Cube of a binomial
- 6 Other products
- 7 Summary of notable products
- 8 Sources
Definition
They are those products that are governed by fixed rules and whose result can be found by simple inspection. Its also called “Algebraic Identities”. They are those products whose development is classic and for this reason it is easily recognized
Square of a binomial
- Binomial squared sum: A binomial squared (sum) is equal to the square of the first term, plus the double product of the first by the second plus the second square.
(a + b) 2 = a2 + 2 • a • b + b2 Example: (x + 3) 2 = x 2 + 2 • x • 3 + 3 2 = x 2 + 6 x + 9
- Binomial of squared subtraction: A binomial squared (subtraction) is equal is equal to the square of the first term, minus the double product of the first by the second, plus the second square.
(a – b) 2 = a2 – 2 • a • b + b2
Example: (2x – 3) 2 = (2x) 2 – 2 • 2x • 3 + 3 2 = 4×2 – 12 x + 9
Conjugated binomials
Conjugated binomials or sum for difference
- A sum for difference equals difference of squares.
(a + b) • (a – b) = a2 – b2
Example: (2x + 5) • (2x – 5) = (2 x) 2 – 52 = 4×2 – 25
Binomials with common term
- Product of two binomials that have a common term
(x + a) (x + b) = x2 + (a + b) x + ab
Example: (x + 2) (x + 3) = x2 + (2 + 3) x + 2 • 3 = x2 + 5x + 6
Cube of a binomial
Binomial cube or Binomial cube
- Binomial of cube sum: A binomial of cube (sum) is equal to the cube of the first, plus three times the square of the first by the second, plus three times the first by the square of the second, plus the cube of the second.
(a + b) 3 = a3 + 3 • a2 • b + 3 • a • b2 + b3 Example (x + 3) 3 = x 3 + 3 • x2 • 3 + 3 • x • 32 + 33 = x 3 + 9×2 + 27x + 27
- Cube binomial subtraction: A cube binomial (subtraction) is equal to the cube of the first, minus three times the square of the first by the second, plus three times the first by the square of the second, minus the cube of the second.
(a – b) 3 = a3 – 3 • a2 • b + 3 • a • b2 – b Example: (2x – 3) 3 = (2x) 3 – 3 • (2x) 2 • 3 + 3 • 2x • 32 – 33 = 8x 3 – 36 x2 + 54 x – 27
Other products
- Trinomial squared: A trinomial squared equals the square of the first, plus the square of the second, plus the square of the third, plus twice the first for the second, plus double the first for the third, plus double the second by the third party.
(a + b + c) 2 = a2 + b2 + c2 + 2 • a • b + + 2 • a • c + 2 • b • c Example: (x2 – x + 1) 2 = (x2) 2 + ( −x) 2 + 12 +2 • x2 • (−x) + 2 x2 • 1 + 2 • (−x) • 1 = x4 + x2 + 1 – 2×3 + 2×2 – 2x = x4 – 2×3 + 3×2 – 2x + one
- Sum of cubes
a3 + b3 = (a + b) • (a2 – ab + b2) Example; 8×3 + 27 = (2x + 3) (4×2 – 6x + 9)
- Cube difference
a3 – b3 = (a – b) • (a2 + ab + b2) 8×3 – 27 = (2x – 3) (4×2 + 6x + 9)
Summary of notable products
- b) (a2 + b2) Fourth difference (a + b + c) 2 = a2 + b2 + c2 + 2ab + 2ac + 2bc Trinomial squared Each notable product corresponds to a factorization formula.- b4 = (a + b) (a – ab) Sum of cubes a4 – b) (a2 + b2 + ab) Difference of cubes a3 + b3 = (a + b) (a2 + b2 – b3 = (a – b) Difference of squares a3 – b2 = (a + b) (a -As a summary, the following table is delivered with notable Products: (a + b) 2 = a2 + 2ab + b2 Binomial squared (a + b) 3 = a3 + 3a2b + 3ab2 + b3 Binomial cube a2 For example, factoring a difference of perfect squares is a product of two binomials conjugated and reciprocally
