## Algebraic Identities Of Polynomials

You can also read https://www.aplustopper.com/ncert-solutions-for-class-10-maths-chapter-2/ for more solved examples.

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## Algebraic Identities Of Polynomials Example Problems With Solutions

**Example 1: **Expand each of the following

**Solution: **(i) We have,

**Example 2: **Find the products

(i) (2x + 3y) (2x – 3y)

**Solution: **(i) We have,

**Example 3: **Evaluate each of the following by using identities

(i) 103 × 97 (ii) 103 × 103

(iii) (97)^{2} (iv) 185 × 185 – 115 × 115

**Solution: **(i) We have,

**Example 4: **

**Solution: **We have,

**Example 5: **

**Solution: **We have,

**Example 6: **If x + y = 12 and xy = 32, find the value of x^{2} + y^{2}

**Solution: **We have,

**Example 7: **Prove that:

2a^{2} + 2b^{2} + 2c^{2} – 2ab – 2bc – 2ca = [(a – b)^{2} + (b – c)^{2} + (c – a)^{2}]

**Solution: **We have,

**Example 8: **If a^{2} + b^{2} + c^{2} – ab – bc – ca = 0, prove that a = b = c.

**Solution: **We have,

**Example 9: **Write the following in expanded form :

(i) (9x + 2y + z)^{2} (ii) (3x + 2y – z)^{2}

(iii) (x – 2y – 3z)^{2} (iv) (–x + 2y + z)^{2}

**Solution: **Using the identity

**Example 10: **If a^{2} + b^{2} + c^{2} = 20 and a + b + c = 0, find ab + bc + ca.

**Solution: **We have,

**Example 11: **If a + b + c = 9 and ab + bc + ca = 40, find a^{2} + b^{2} + c^{2}.

**Solution: **We know that

**Example 12: **If a^{2} + b^{2} + c^{2} = 250 and ab + bc + ca = 3, find a + b + c.

**Solution: **We know that

**Example 13: **Write each of the following in expanded form:

(i) (2x + 3y)^{3} (ii) (3x – 2y)^{3}

**Solution: **(i)** **Replacing a by 2x and b by 3y in the identity

**Example 14: ** If x + y = 12 and xy = 27, find the value of x^{3} + y^{3}.

**Solution: **We know that

**Example 15: **If x – y = 4 and xy = 21, find the value of x^{3} – y^{3}.

**Solution: **We know that

**Example 16: **

**Solution: **We have,

**Example 17: **If a + b = 10 and a2 + b2 = 58, find the value of a^{3} + b^{3}.

**Solution: **We know that

**Example 18: **

**Solution: **We have,

**Example 19: **

**Solution: **We know that

**Example 20: **If a + b = 10 and ab = 21, find the value of a^{3} + b^{3}.

**Solution: **We know that

**Example 21: **If a – b = 4 and ab = 45, find the value of a^{3} – b^{3}.

**Solution: **We have,

**Example 22: ** If a + b + c = 0, then prove that a3 + b3 + c3 = 3abc ** **

**Solution: **We know that

**Example 23: **Find the following product:

(x + y + 2z) (x^{2} + y^{2} + 4z^{2} – xy – 2yz – 2zx)

**Solution: **We have,

**Example 24: ** If a + b + c = 6 and ab + bc + ca = 11, find the value of a^{3} + b^{3} + c^{3} – 3abc.

**Solution: **We know that

**Example 25: **If x + y + z = 1, xy + yz + zx = –1 and xyz = –1, find the value of x^{3} + y^{3} + z^{3}.

**Solution: **We know that

pranav says

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Ammrutha says

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Rishika Raj says

So helpful for me in mathematics algebraic identities

Devansh judo says

It is very nice and it is helpful also

Prateek says

They are very helpful in exams

Divyanshu shukla says

Tomorrow Is my exam of maths thanks for uploading it

ansh tiwari says

what is the formula of

(a+b)whole cube – (a-b)whole cube

Anon says

6a^2 + 2b^3

nisarga says

i am not having exams or tests on tomorrow just i was searhing some new identities i got this and it helped me much

Sipun says

It’s realy very helpful

Karam says

Very useful.