## Problems on identities In this page 'Problems on identities' we are going to see problems on identities.

Parents and teachers can guide the students to do the problems on their own. If they are having any doubt they can verify the solutions.

Before going to the problems let us recall about the identities.

1.(a + b)² =a² + 2 ab + b²

2.(a - b)²  = a² -  2 ab  +  b²

3.  a²-b²     =  (a+b)(a-b)

4.(x+a)(x+b) = x² + (a+b)x + ab

5. (x+y+z)²  =  x² + y² + z² + 2xy + 2yz + 2zx

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

7. (x+y)³     =  x³ +3x²y +3xy² + y³  = x³ + y³ +3xy(x+y)

8. (x-y)³     =  x³ - 3x²y +3xy² - y³   = x³ - y³ - 3xy(x-y)

9. x³ + y³ + z³ -3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx)

Note:

If x+y+z = 0, then x³ + y³ +z³ = 3xyz

### Problems on identities

1.     Expand the following

(i)     (5x + 2y + 3z)²

(ii)      (2a + 3b - c)²

(iii)     (x-2y-4z )²

(iv)     (p - 2q + r)²

2.      Find the expansion of

(i)     (x+1)(x+4)(x+7)

(ii)    (p+2)(p-4)(p+6)

(iii)    (x+5)(x-3)(x-1)

(iv)    (x-a)(x-2a)(x-4a)

(v)     (3x+1)(3x+2)((3x+5)

(vi)     (2x+3)(2x-5)(2x-7)

3.       Using algebraic identities find the coefficients of x² term, x term and constant terms.

(i)       (x+7)(x+3)(x+9)

(ii)      (x-5)(x-4)(x+2)

(iii)     (2x+3)(2x+5)(2x+7)

(iv)     (5x+2)(1-5x)(5x+3)

4.         If (x+a)(x+b)(x+c) ≡ x³-10x² + 45x -15 find a+b+c, 1/a+1/b+1/c and

a²+b²+c².

5.        Expand:

(i)        (3a+5b)³

(ii)        (4x-3y)³

(iii)       (2y - 3/y)³

6.         Evaluate:

(i)        99³

(ii)        101³

(iii)        98³

(iv)       102³

(v)        1002³

7.         Find 8x³ + 27y³ if 2x + 3y = 13 and xy = 16.

8.         If x-y = -6 and xy = 4, find the value of x³ - y³.

9.         If  x + 1/x = 4, find the value of x³ + 1/x³.

10.      If x - 1/x = 3, find the value of x³ - 1/x³

11.      Simplify:

(i)       (2x+y+4z)(4x²+y²+16z²-2xy -4yz -8zx)

(ii)       (x-3y-5z)(x²+9y²+25z²+3xy-15yz+5zx)

12.      Evaluate using identities:

(i)        6³-9³+3³

(ii)        16³-6³-10³

Solutions

Students can try to solve the problems given above in 'problems on algebraic identities', on their own. Parents and teachers can encourage the students to do so. If they are having any doubt they can verify the solutions.  If you are having any further doubt you can contact us through mail, we will help you to clear your doubt.