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# Identities, Cube of a binomial, Algebra, Math Assignment Help

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mathematics - Identities, Cube of a binomial, Algebra, Math

**Identity:** An identity is an equality which is true for all values of variables.

**IMPORTANT FORMULAE**

(1) To Find the Product (x + a) (x + b): (x + a) (x + b)

= x^{2} + (a + b)x + ab

Using this identity, we can derive the following results:

When b = a, we have

(x + a) (x + a) = x^{2} + 2ax + a^{2}

When = - a, we have = x^{2} - a^{2} When b = - b, we have

= x^{2} + (a - b)x - ab

When a = - a, we have = x^{2} + (b - a) x - ab

When a = - a, b = - b, we have = x^{2} - (a + b) x + ab.

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a - b)^{2} = a^{2} - 2ab + b^{2}

(a + b) (a - b) = a^{2} - b^{2}.

To Find the Product (a + b) (a^{2} - ab + b^{2}): (a + b) (a^{2} - ab + b^{2})

= a^{3} + b^{3} To Find the Product (a - b)(a^{2} + ab + b^{2}) :

(a - b) (a^{2} + ab + b^{2}) = a^{3} - b^{3}

To Find the Product (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac):

(a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac) = a^{3} + b^{3} + c^{3} - 3abc.

(2) **Cube of a binomial: **

(a + b)^{3} = a^{3} + b^{3} + 3ab (a + b)

a^{3} + b^{3} = (a + b)^{3} - 3ab (a + b)

(a - b)^{3} = a^{3} - b^{3} - 3ab (a - b)

a^{3} - b^{3} = (a - b)^{3} + 3ab (a - b).

(3) **Square of a trinomial: **

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac.

**Note:** a, b, c may take various positive and negative values:

(i) (- a - b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab - 2ac - 2bc.

(ii) (a - b + c)^{2} = a^{2} + b^{2} + c^{2} - 2ab - 2bc + 2ac.

(iii) (- a + b + c)^{2} = a^{2} + b^{2} + c^{2} - 2ab + 2bc - 2ac.

(iv) (a - b - c)^{2} = a^{2} + b^{2} + c^{2} - 2ab + 2bc + 2ac.

(v) (a + b - c)^{2} = a^{2} + b^{2} + c^{2} + 2ab - 2bc - 2ac.

**An important result:** If a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc.

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**Identity:**An identity is an equality which is true for all values of variables.

**IMPORTANT FORMULAE**

^{2}+ (a + b)x + ab

^{2}+ 2ax + a

^{2}

^{2}- a

^{2}When b = - b, we have

^{2}+ (a - b)x - ab

^{2}+ (b - a) x - ab

^{2}- (a + b) x + ab.

^{2}= a

^{2}+ 2ab + b

^{2}

^{2}= a

^{2}- 2ab + b

^{2}

^{2}- b

^{2}.

^{2}- ab + b

^{2}): (a + b) (a

^{2}- ab + b

^{2})

^{3}+ b

^{3}To Find the Product (a - b)(a

^{2}+ ab + b

^{2}) :

^{2}+ ab + b

^{2}) = a

^{3}- b

^{3}

^{2}+ b

^{2}+ c

^{2}- ab - bc - ac):

^{2}+ b

^{2}+ c

^{2}- ab - bc - ac) = a

^{3}+ b

^{3}+ c

^{3}- 3abc.

**Cube of a binomial:**

^{3}= a

^{3}+ b

^{3}+ 3ab (a + b)

^{3}+ b

^{3}= (a + b)

^{3}- 3ab (a + b)

^{3}= a

^{3}- b

^{3}- 3ab (a - b)

^{3}- b

^{3}= (a - b)

^{3}+ 3ab (a - b).

**Square of a trinomial:**

^{2}= a

^{2}+ b

^{2}+ c

^{2}+ 2ab + 2bc + 2ac.

**Note:**a, b, c may take various positive and negative values:

^{2}= a

^{2}+ b

^{2}+ c

^{2}+ 2ab - 2ac - 2bc.

^{2}= a

^{2}+ b

^{2}+ c

^{2}- 2ab - 2bc + 2ac.

^{2}= a

^{2}+ b

^{2}+ c

^{2}- 2ab + 2bc - 2ac.

^{2}= a

^{2}+ b

^{2}+ c

^{2}- 2ab + 2bc + 2ac.

^{2}= a

^{2}+ b

^{2}+ c

^{2}+ 2ab - 2bc - 2ac.

**An important result:**If a + b + c = 0, then a

^{3}+ b

^{3}+ c

^{3}= 3abc.