CBSE Class 8 Maths Algebraic Expressions and Identities HOTs

CHALLENGES

1. Use appropriate formulae and compute:
a. (103)² b. (96)² c. 107 × 93
d. 1008 × 992 e. 185² – 115²

2. If x+y=7 and xy=12, find x²+y².

3. If x+y=12 and xy=32, find x²+y².

4. If 4x²+y²=40 and xy=6, find 2x+y.

5. If x–y=3 and xy=10, find x²+y².

6. If x+ 1/x = 3, find x²+1/x² and x³+ 1/x³ .

8. Simplify:
a. (x+y)²+(x–y)² b. (x+y)²×(x–y)².

9. Express the following as difference of two squares:
a. (x+2z)(2x+z) b. 4(x+2y)(2x+y)
c. (x+98)(x+102) d. 505×495
(Hint : ab=(a+b/2)² - (a-b/2)²

10. If A=3x–5y, B=6x+3y and C=2y–4x, find
a. A+B–C b. 2A–3B+4C

11. The perimeter of a triangle is 15x²–23x+9 and two of its sides are 5x2+8x–1 and 6x2–9x+4. Find the third side.

12. The two adjacent sides of a rectangle are 2x²–5xy+3z² and 4xy–x²–z2. Find its perimeter.

13. The base and the altitude of a triangle are (3x–4y) and (6x+5y) respectively. Find its area.

14. The sides of a rectangle are 2x+3y and 3x+2y. From this a square of side length x+y is removed.
What is the area of the remaining region?

15. If a, b, c are rational numbers such that a²+b²+c²–ab–bc–ca=0, prove that a=b=c.

16. Express a²+b² using a+b and a–b.

17. Express (a²+b²)(c²+d²) as the sum of the squares of two algebraic expressions.

18. Prove that (x³+y³)²–(x²+y²)³+3x²y²(x+y)²= (2xy)³.

19. Prove that (x–y)²+(y–z)²+(z–x)²=2(x–y)(x–z)+2(y–z)(y–x)+2(z–x)(z–y).

SUMMARY

1. A combination of constants, variables and basic mathematical operators i.e. +, –, ×, ÷. is an algebraic expression.

2. An expression with non-negative integral exponents of variables is called a polynomial.

3. On the basis of number of terms, polynomials may be monomial, binomial and trinomial.

4. Highest exponent of the variable in an expression is its degree.

5. Basic identities are :
(a+b)² = a²+2ab+b²
(a –b)² - a²–2ab+b²
(a+b)(a–b) = a²–b²
(x+a) (x+b) = x²+(a+b)x+ab

ERRORANALYSIS

1. Students make a lot of calculation mistakes.
2. Students fail to memorize the identities and do not apply them correctly.
3. Some of the common errors while handling the expressions are :
a. 3x+x+5x=8x [As coefficient ‘1’ of the 2nd term is not mentioned, students fail to notice it]
3x+x+5x=9x [correct method]
b. 3(x–4) =3x–4 [Students forget that 3 is also to be multiplied with –4]
3(x–4) = 3x+3(–4) [Correct method]
=3x–12
c. (2x)²=2x² [Students miss to square the numerical coefficient]
(2x)²=4x² [Correct method]
d. (2x+3)² = 4x²+9 [Incorrect application of identity]
(2x+3)² = 4x²+2(2x)(3)+9 [Correct method]
= 4x²+12x+9

ACTIVITY

Algebraic Tiles

Cut the following tiles from a graph sheet. Now, colour the tiles as per the colour code. Arrange these algebraic tiles to form a square. 

a. Find the length of the side of the square so formed. Also find the area of the square. Using the above result, factorise x²+4x+4. 

b. Find the length of the side of the rectangle so formed. Also find the area of the rectangle. Using the above result, factorise x²+5x+4.
Now choose and cut more algebraic tiles from the graph sheet. Create your own colour code and colour the tiles. Arrange them to form square/rectangle. Find the area of the figure so formed. Use it to factorise
a) x²+4x+3 b) x²+9x+18

Very Short Answer type Questions

Question. Simplify 4m² + 9n² - 12mn / 2m - 3n
Answer : 2m – 3n

Question. Simplify : (3m + 2n)² – (9m² + 4n²).
Answer : 12mn

Question. Find the value of (3m + 5n)² – 60mn.
Answer : (3m – 5n)²

Question. Find the value of (9p – 5q)² + 180pq
Answer : (9p + 5q)²

Question. Find the volume of a rectangular box with length, breadth and height respectively 3ab, 4bc and 5ca.
Answer : 60a²b²c²

Question. Simplify : (p² + 5) (p – 5) + 25.
Answer : p³ – 5p² + 5p

Question. Find the value of x if 3x – 9b = 3b.
Answer : 4b

Question. Find the square of (3xy + 4yz).
Answer : 9x²y² + 24xy²z + 16y²z²

Question. Find the product of (3m + 2) (3m + 3).
Answer : 9m² + 15m + 6

Question. Simplify : (3m²np)²
Answer : 9m⁴n²p²

Question. Identify the coefficients of the algebric expression 2a²by – 3ay.
Answer : (2, –3)

Question. Find the product of (x + y) (x – y) (x² + y²).
Answer : x⁴ – y⁴

Question. Simplify : (x² + 4) (x² – 4) + 16.
Answer : x⁴

Question. How much is (√13 + √11) (√13 – √11)
Answer : 2

Question. Subtract : 3pq (p – q) from 4pq (p – q).
Answer : p²q – pq²