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**

- a
^{2}− b^{2}= ( a + b) (a − b) - (a + b)
^{2 }= a^{2}+ 2ab + b^{2} - a
^{2}+ b^{2 }= (a − b)^{2}+ 2ab - (a − b)
^{2}= a^{2}− 2ab + b^{2} - (a + b + c)
^{2 }= a^{2}+ b^{2}+ c^{2}+ 2ab + 2ac + 2bc - (a − b − c)
^{2}= a^{2}+ b^{2 }+ c^{2 }− 2ab − 2ac + 2bc - (a + b)
^{3}= a^{3 }+ 3a^{2}b + 3ab^{2}+ b^{3} - (a + b)
^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a − b)
^{3}= a^{3}− 3a^{2}b + 3ab^{2 }− b^{3} - a
^{3}− b^{3}= (a − b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b) (a^{2}− ab + b^{2}) - (a + b)
^{4}= a^{4}+ 4a^{3}b + 6a^{2}b^{2}+ 4ab^{3}+ b^{4} - (a − b)
^{4}= a^{4}− 4a^{3}b + 6a^{2}b^{2}− 4ab^{3}+ b^{4} - a
^{4}−b^{4}= (a − b)(a + b)(a^{2}+ b^{2}) - a
^{5}−b^{5}=(a − b)(a^{4}+ a^{3}b + a^{2}b^{2}+ ab^{3}+ b^{4}) - (x + y + z)
^{2}= x^{2 }+ y^{2 }+ z^{2 }+ 2xy + 2yz + 2xz - (x + y − z)
^{2}= x^{2}+ y^{2}+ z^{2}+ 2xy − 2yz − 2xz - (x − y + z)
^{2 }= x^{2}+ y^{2}+ z^{2}− 2xy − 2yz + 2xz - (x − y − z)
^{2}= x^{2}+ y^{2}+ z^{2}− 2xy + 2yz − 2xz - x
^{3}+ y^{3}+ z^{3}−3xyz = (x + y + z)(x^{2}+ y^{2 }+ z^{2 }− xy − yz − xz) - x
^{2}+ y^{2 }= 1/2 [(x + y)^{2 }+ (x − y)^{2}] - (x + a)(x + b)(x + c) = x
^{3 }+ (a + b + c)x^{2 }+ (ab + bc + ca)x + abc - x
^{3}+ y^{3}= (x + y)(x^{2}− xy + y^{2}) - x
^{3}− y^{3}= (x − y)(x^{2}+ xy + y^{2}) - x
^{2}+ y^{2 }+ z^{2 }− xy − yz − zx = 1/2[(x − y)^{2}+ (y − z)^{2}+ (z − x)^{2}] **If n is a natural number,**a^{n}− b^{n}= (a − b)(a^{n−1}+ a^{n−2}b + … + b^{n−2}a + b^{n−1})**If n is even**(n = 2k), a^{n}+ b^{n }= (a + b)(a^{n−1}− a^{n−2}b + … + b^{n−2}a − b^{n−1})**If n is odd**(n = 2k + 1), a^{n}+ b^{n}= (a + b)(a^{n−1}− a^{n−2}b + … − b^{n−2}a + b^{n−1})- (a + b + c + …)
^{2 }= a^{2 }+ b^{2 }+ c^{2 }+ … + 2(ab + bc + ….) **Laws of Exponents**- (a
^{m})(a^{n}) = a^{m+n} - (ab)
^{m}= a^{m}b^{m} - (a
^{m})^{n}= a^{mn}

- (a
**Fractional Exponents**- a
^{0}= 1 - a
^{m}/a^{n}= a^{m−n} - a
^{m}= 1/a^{−m} - a
^{−m}= 1/a^{m}

- a

**Practice Problem:**

**Find value of (3 + 7)**^{2}

**Solution:** Using formula (a + b)^{2} = a^{2} + b^{2} + 2ab

(3 + 7)^{2} = 3^{2} + 7^{2} + 2(3)(7)

= 9 + 49 + 42

= 100

**Find the value of (5 + 4 − 3)**^{2}

**Solution: **Using formula (a+b+c)^{2}=a^{2} + b^{2} + c^{2} + 2ab + 2ac + 2bc

= (5 + 4 + (-3))^{2} = 5^{2} + 4^{2} + (-3)^{2} + 2*5*4 + 2*5*(-3) + 2*4*(-3)

= 25 + 16 + 9 + 40 – 30 – 12

= 48