Algebra Formula

Algebra is a branch of Mathematics that substitutes letters for numbers. The numbers are constants. Algebra also includes real numbers, complex numbers, matrices, vectors and much more.  X, Y, A, B are the most commonly used letters that represent the algebraic problems and equation 1/b.

Important Formulas in Algebra

Real Numbers : a, b, c, x, y, z
Natural Number : n, m

Factoring Formulas

• a2 − b2 = ( a + b) (a − b)

• (a + b)2 = a2 + 2ab + b2

• a2 + b2 = (a − b)2 + 2ab

• (a − b)2 = a2 − 2ab + b2

• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc

• (a − b − c)2 = a2 + b2 + c2 − 2ab − 2ac + 2bc

• (a + b)3 = a3 + 3a2b + 3ab2 + b3

• (a + b)3 = a3 + b3 + 3ab(a + b)

• (a − b)3= a3 − 3a2b + 3ab2 − b3

• a3 − b3 = (a − b)(a2 + ab + b2)

• a3 + b3 = (a + b) (a2 − ab + b2)

• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

• (a − b)4 = a4 − 4a3b + 6a2b2 − 4ab3 + b4

• a4 −b4 = (a − b)(a + b)(a2 + b2)

• a5 −b5 =(a − b)(a4 + a3b + a2b2 + ab3 + b4)

• (x + y + z)2= x2 + y2 + z2 + 2xy + 2yz + 2xz

• (x + y − z)2= x2 + y2 + z2 + 2xy − 2yz − 2xz

• (x − y + z)2 = x2 + y2 + z2 − 2xy − 2yz + 2xz

• (x − y − z)2= x2 + y2 + z2 − 2xy + 2yz − 2xz

• x3 + y3 + z3 −3xyz = (x + y + z)(x2 + y2 + z2 − xy − yz − xz)

• x2 + y2 = 1/2 [(x + y)2 + (x − y)2]

• (x + a)(x + b)(x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc

• x3 + y3 = (x + y)(x2 − xy + y2)

• x3 − y3 = (x − y)(x2 + xy + y2)

• x2 + y2 + z2 − xy − yz − zx = 1/2[(x − y)2 + (y − z)2 + (z − x)2]

• If n is a natural number, an − bn = (a − b)(an−1 + an−2b + … + bn−2a + bn−1)

• If n is even (n = 2k), an + bn = (a + b)(an−1 − an−2b + … + bn−2a − bn−1)

• If n is odd (n = 2k + 1), an + bn = (a + b)(an−1 − an−2b + … − bn−2a + bn−1)

• (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + bc + ….)

• Laws of Exponents
• (am)(an) = am+n
• (ab)m = ambm
• (am)n = amn

• Fractional Exponents
• a0 = 1
• am/an = am−n
• am = 1/a−m
• a−m = 1/am

Practice Problem:

1. Find value of (3 + 7)2

Solution: Using formula (a + b)2 = a2 + b2 + 2ab

(3 + 7)2 = 32 + 72 + 2(3)(7)

= 9 + 49 + 42

= 100

1. Find the value of (5 + 4 − 3)2

Solution: Using formula (a+b+c)2=a2 + b2 + c2 + 2ab + 2ac + 2bc

= (5 + 4 + (-3))2 = 52 + 42 + (-3)2 + 2*5*4 + 2*5*(-3) + 2*4*(-3)

= 25 + 16 + 9 + 40 – 30 – 12

= 48