Chapter 12: Algebraic Expressions (NCERT Solutions)
Exercise 12.1
(i) Subtraction of z from y.
(ii) One-half of the sum of numbers x and y.
(iii) The number z multiplied by itself.
(iv) One-fourth of the product of numbers p and q.
(v) Numbers x and y both squared and added.
(vi) Number 5 added to three times the product of numbers m and n.
(vii) Product of numbers y and z subtracted from 10.
(viii) Sum of numbers a and b subtracted from their product.
(i) y - z
(ii) ½(x + y) or (x + y)/2
(iii) z × z = z²
(iv) ¼(pq) or pq/4
(v) x² + y²
(vi) 3mn + 5
(vii) 10 - yz
(viii) ab - (a + b)
(a) x - 3
(b) 1 + x + x²
(c) y - y³
(d) 5xy² + 7x²y
(e) -ab + 2b² - 3a²
(a) x - 3
Terms: x and -3
Factors of x: x
Factors of -3: -3 (or -1, 3)
(b) 1 + x + x²
Terms: 1, x, and x²
Factors of 1: 1
Factors of x: x
Factors of x²: x, x
(c) y - y³
Terms: y and -y³
Factors of y: y
Factors of -y³: -1, y, y, y
(d) 5xy² + 7x²y
Terms: 5xy² and 7x²y
Factors of 5xy²: 5, x, y, y
Factors of 7x²y: 7, x, x, y
(e) -ab + 2b² - 3a²
Terms: -ab, 2b², and -3a²
Factors of -ab: -1, a, b
Factors of 2b²: 2, b, b
Factors of -3a²: -3, a, a (or -1, 3, a, a)
(a) -4x + 5
(b) -4x + 5y
(c) 5y + 3y²
(d) xy + 2x²y²
(e) pq + q
(f) 1.2ab - 2.4b + 3.6a
(g) ¾x + ¼
(h) 0.1p² + 0.2q²
| Expression | Terms | Factors |
|---|---|---|
| (a) -4x + 5 | -4x 5 | -4, x 5 |
| (b) -4x + 5y | -4x 5y | -4, x 5, y |
| (c) 5y + 3y² | 5y 3y² | 5, y 3, y, y |
| (d) xy + 2x²y² | xy 2x²y² | x, y 2, x, x, y, y |
| (e) pq + q | pq q | p, q q |
| (f) 1.2ab - 2.4b + 3.6a | 1.2ab -2.4b 3.6a | 1.2, a, b -2.4, b 3.6, a |
| (g) ¾x + ¼ | ¾x ¼ | ¾, x ¼ |
| (h) 0.1p² + 0.2q² | 0.1p² 0.2q² | 0.1, p, p 0.2, q, q |
(i) 5 - 3t² (ii) 1 + t + t² + t³ (iii) x + 2xy + 3y
(iv) 100m + 1000n (v) -p²q² + 7pq (vi) 1.2a + 0.8b
(vii) 3.14r² (viii) 2(l + b) (ix) 0.1y + 0.01y²
(i) 5 - 3t²
Term: -3t² ⇒ Coefficient: -3
(ii) 1 + t + t² + t³
Term: t ⇒ Coefficient: 1
Term: t² ⇒ Coefficient: 1
Term: t³ ⇒ Coefficient: 1
(iii) x + 2xy + 3y
Term: x ⇒ Coefficient: 1
Term: 2xy ⇒ Coefficient: 2
Term: 3y ⇒ Coefficient: 3
(iv) 100m + 1000n
Term: 100m ⇒ Coefficient: 100
Term: 1000n ⇒ Coefficient: 1000
(v) -p²q² + 7pq
Term: -p²q² ⇒ Coefficient: -1
Term: 7pq ⇒ Coefficient: 7
(vi) 1.2a + 0.8b
Term: 1.2a ⇒ Coefficient: 1.2
Term: 0.8b ⇒ Coefficient: 0.8
(vii) 3.14r²
Term: 3.14r² ⇒ Coefficient: 3.14
(viii) 2(l + b) = 2l + 2b
Term: 2l ⇒ Coefficient: 2
Term: 2b ⇒ Coefficient: 2
(ix) 0.1y + 0.01y²
Term: 0.1y ⇒ Coefficient: 0.1
Term: 0.01y² ⇒ Coefficient: 0.01
(i) y²x + y (ii) 13y² - 8yx (iii) x + y + 2
(iv) 5 + z + zx (v) 1 + x + xy (vi) 12xy² + 25
(vii) 7x + xy²
| Expression | Term with x | Coefficient of x |
|---|---|---|
| y²x + y | y²x | y² |
| 13y² - 8yx | -8yx | -8y |
| x + y + 2 | x | 1 |
| 5 + z + zx | zx | z |
| 1 + x + xy | x xy | 1 y |
| 12xy² + 25 | 12xy² | 12y² |
| 7x + xy² | 7x xy² | 7 y² |
(i) 8 - xy² (ii) 5y² + 7x (iii) 2x²y - 15xy² + 7y²
| Expression | Term with y² | Coefficient of y² |
|---|---|---|
| 8 - xy² | -xy² | -x |
| 5y² + 7x | 5y² | 5 |
| 2x²y - 15xy² + 7y² | -15xy² 7y² | -15x 7 |
(i) 4y - 7z (ii) y² (iii) x + y - xy
(iv) 100 (v) ab - a - b (vi) 5 - 3t
(vii) 4p²q - 4pq² (viii) 7mn (ix) z² - 3z + 8
(x) a² + b² (xi) z² + z (xii) 1 + x + x²
Monomials (1 term): (ii) y², (iv) 100, (viii) 7mn
Binomials (2 terms): (i) 4y - 7z, (vi) 5 - 3t, (vii) 4p²q - 4pq², (x) a² + b², (xi) z² + z
Trinomials (3 terms): (iii) x + y - xy, (v) ab - a - b, (ix) z² - 3z + 8, (xii) 1 + x + x²
(i) 1, 100 (ii) -7x, 5/2x (iii) -29x, -29y
(iv) 14xy, 42yx (v) 4m²p, 4mp² (vi) 12xz, 12x²z²
(i) 1, 100: Like terms (Both are constants)
(ii) -7x, 5/2x: Like terms (Same variable factor x)
(iii) -29x, -29y: Unlike terms (Different variables x and y)
(iv) 14xy, 42yx: Like terms (xy is same as yx)
(v) 4m²p, 4mp²: Unlike terms (Variables are same but powers are different, m² ≠ m)
(vi) 12xz, 12x²z²: Unlike terms (Powers of variables are different)
(a) -xy², -4yx², 8x², 2xy², 7y, -11x², -100x, -11yx, 20x²y, -6x², y, 2xy, 3x
(b) 10pq, 7p, 8q, -p²q², -7qp, -100q, -23, 12q²p², -5p², 41, 2405p, 78qp, 13p²q, qp², 701p²
(a) Groups of like terms:
1. -xy², 2xy²
2. -4yx², 20x²y
3. 8x², -11x², -6x²
4. 7y, y
5. -100x, 3x
6. -11yx, 2xy
(b) Groups of like terms:
1. 10pq, -7qp, 78qp
2. 7p, 2405p
3. 8q, -100q
4. -p²q², 12q²p²
5. -23, 41
6. -5p², 701p²
7. 13p²q, qp²
Exercise 12.2
(i) 21b - 32 + 7b - 20b
(ii) -z² + 13z² - 5z + 7z³ - 15z
(iii) p - (p - q) - q - (q - p)
(iv) 3a - 2b - ab - (a - b + ab) + 3ab + b - a
(v) 5x²y - 5x² + 3yx² - 3y² + x² - y² + 8xy² - 3y²
(vi) (3y² + 5y - 4) - (8y - y² - 4)
(i) 21b - 32 + 7b - 20b
= (21b + 7b - 20b) - 32
= b(21 + 7 - 20) - 32
= 8b - 32
(ii) -z² + 13z² - 5z + 7z³ - 15z
= 7z³ + (-z² + 13z²) + (-5z - 15z)
= 7z³ + 12z² - 20z
(iii) p - (p - q) - q - (q - p)
= p - p + q - q - q + p
= (p - p + p) + (q - q - q)
= p - q
(iv) 3a - 2b - ab - (a - b + ab) + 3ab + b - a
= 3a - 2b - ab - a + b - ab + 3ab + b - a
= (3a - a - a) + (-2b + b + b) + (-ab - ab + 3ab)
= a + ab
(v) 5x²y - 5x² + 3yx² - 3y² + x² - y² + 8xy² -
3y²
Note: 3yx² is same as 3x²y.
= (5x²y + 3x²y) + 8xy² + (-5x² + x²) + (-3y² - y² -
3y²)
= 8x²y + 8xy² - 4x² - 7y²
(vi) (3y² + 5y - 4) - (8y - y² - 4)
= 3y² + 5y - 4 - 8y + y² + 4
= (3y² + y²) + (5y - 8y) + (-4 + 4)
= 4y² - 3y
(i) 3mn, -5mn, 8mn, -4mn
(ii) t - 8tz, 3tz - z, z - t
(iii) -7mn + 5, 12mn + 2, 9mn - 8, -2mn - 3
(iv) a + b - 3, b - a + 3, a - b + 3
(v) 14x + 10y - 12xy - 13, 18 - 7x - 10y + 8xy, 4xy
(vi) 5m - 7n, 3n - 4m + 2, 2m - 3mn - 5
(vii) 4x²y, -3xy², -5xy², 5x²y
(i) 3mn, -5mn, 8mn, -4mn
= 3mn - 5mn + 8mn - 4mn
= (3 - 5 + 8 - 4)mn = 2mn
(ii) t - 8tz, 3tz - z, z - t
= t - 8tz + 3tz - z + z - t
= (t - t) + (-8tz + 3tz) + (-z + z)
= -5tz
(iii) -7mn + 5, 12mn + 2, 9mn - 8, -2mn - 3
= -7mn + 5 + 12mn + 2 + 9mn - 8 - 2mn - 3
= (-7mn + 12mn + 9mn - 2mn) + (5 + 2 - 8 - 3)
= 12mn - 4
(iv) a + b - 3, b - a + 3, a - b + 3
= a + b - 3 + b - a + 3 + a - b + 3
= (a - a + a) + (b + b - b) + (-3 + 3 + 3)
= a + b + 3
(v) 14x + 10y - 12xy - 13, 18 - 7x - 10y + 8xy, 4xy
= 14x + 10y - 12xy - 13 + 18 - 7x - 10y + 8xy + 4xy
= (14x - 7x) + (10y - 10y) + (-12xy + 8xy + 4xy) + (-13 + 18)
= 7x + 0 + 0 + 5 = 7x + 5
(vi) 5m - 7n, 3n - 4m + 2, 2m - 3mn - 5
= 5m - 7n + 3n - 4m + 2 + 2m - 3mn - 5
= (5m - 4m + 2m) + (-7n + 3n) - 3mn + (2 - 5)
= 3m - 4n - 3mn - 3
(vii) 4x²y, -3xy², -5xy², 5x²y
= 4x²y - 3xy² - 5xy² + 5x²y
= (4x²y + 5x²y) + (-3xy² - 5xy²)
= 9x²y - 8xy²
(i) -5y² from y²
(ii) 6xy from -12xy
(iii) (a - b) from (a + b)
(iv) a(b - 5) from b(5 - a)
(v) -m² + 5mn from 4m² - 3mn + 8
(vi) -x² + 10x - 5 from 5x - 10
(vii) 5a² - 7ab + 5b² from 3ab - 2a² - 2b²
(viii) 4pq - 5q² - 3p² from 5p² + 3q² - pq
(i) y² - (-5y²)
= y² + 5y² = 6y²
(ii) -12xy - 6xy
= -18xy
(iii) (a + b) - (a - b)
= a + b - a + b = 2b
(iv) b(5 - a) - a(b - 5)
= 5b - ab - (ab - 5a)
= 5b - ab - ab + 5a = 5a + 5b - 2ab
(v) (4m² - 3mn + 8) - (-m² + 5mn)
= 4m² - 3mn + 8 + m² - 5mn
= (4m² + m²) + (-3mn - 5mn) + 8 = 5m² - 8mn + 8
(vi) (5x - 10) - (-x² + 10x - 5)
= 5x - 10 + x² - 10x + 5
= x² + (5x - 10x) + (-10 + 5) = x² - 5x - 5
(vii) (3ab - 2a² - 2b²) - (5a² - 7ab + 5b²)
= 3ab - 2a² - 2b² - 5a² + 7ab - 5b²
= (-2a² - 5a²) + (-2b² - 5b²) + (3ab + 7ab) = -7a² - 7b² +
10ab
(viii) (5p² + 3q² - pq) - (4pq - 5q² - 3p²)
= 5p² + 3q² - pq - 4pq + 5q² + 3p²
= (5p² + 3p²) + (3q² + 5q²) + (-pq - 4pq) = 8p² + 8q² -
5pq
Let the required expression be A.
(x² + xy + y²) + A = 2x² + 3xy
A = (2x² + 3xy) - (x² + xy + y²)
A = 2x² + 3xy - x² - xy - y²
A = (2x² - x²) + (3xy - xy) - y²
A = x² + 2xy - y²
Let the required expression be B.
(2a + 8b + 10) - B = -3a + 7b + 16
B = (2a + 8b + 10) - (-3a + 7b + 16)
B = 2a + 8b + 10 + 3a - 7b - 16
B = (2a + 3a) + (8b - 7b) + (10 - 16)
B = 5a + b - 6
Let the required expression be C.
(3x² - 4y² + 5xy + 20) - C = -x² - y² + 6xy + 20
C = (3x² - 4y² + 5xy + 20) - (-x² - y² + 6xy + 20)
C = 3x² - 4y² + 5xy + 20 + x² + y² - 6xy - 20
C = (3x² + x²) + (-4y² + y²) + (5xy - 6xy) + (20 - 20)
C = 4x² - 3y² - xy
Sum = (3x - y + 11) + (-y - 11)
= 3x - y + 11 - y - 11
= 3x - 2y
Now, subtract (3x - y - 11) from the Sum:
= (3x - 2y) - (3x - y - 11)
= 3x - 2y - 3x + y + 11
= (3x - 3x) + (-2y + y) + 11
= -y + 11
Sum 1 = (4 + 3x) + (5 - 4x + 2x²)
= 4 + 3x + 5 - 4x + 2x²
= 2x² - x + 9
Sum 2 = (3x² - 5x) + (-x² + 2x + 5)
= 3x² - 5x - x² + 2x + 5
= 2x² - 3x + 5
Now, subtract Sum 2 from Sum 1:
= (2x² - x + 9) - (2x² - 3x + 5)
= 2x² - x + 9 - 2x² + 3x - 5
= (2x² - 2x²) + (-x + 3x) + (9 - 5)
= 2x + 4
Exercise 12.3
(i) m - 2 (ii) 3m - 5 (iii) 9 - 5m
(iv) 3m² - 2m - 7 (v) 5m/2 - 4
Substitute m = 2 in each expression:
(i) m - 2 = 2 - 2 = 0
(ii) 3m - 5 = 3(2) - 5 = 6 - 5 = 1
(iii) 9 - 5m = 9 - 5(2) = 9 - 10 = -1
(iv) 3m² - 2m - 7 = 3(2)² - 2(2) - 7
= 3(4) - 4 - 7 = 12 - 4 - 7 = 1
(v) 5m/2 - 4 = 5(2)/2 - 4
= 10/2 - 4 = 5 - 4 = 1
(i) 4p + 7
(ii) -3p² + 4p + 7
(iii) -2p³ - 3p² + 4p + 7
Substitute p = -2 in each expression:
(i) 4p + 7 = 4(-2) + 7 = -8 + 7 = -1
(ii) -3p² + 4p + 7 = -3(-2)² + 4(-2) + 7
= -3(4) - 8 + 7 = -12 - 8 + 7 = -13
(iii) -2p³ - 3p² + 4p + 7 = -2(-2)³ - 3(-2)² + 4(-2) + 7
= -2(-8) - 3(4) - 8 + 7
= 16 - 12 - 8 + 7
= 4 - 8 + 7 = 3
(i) 2x - 7 (ii) -x + 2 (iii) x² + 2x + 1
(iv) 2x² - x - 2
Substitute x = -1 in each expression:
(i) 2x - 7 = 2(-1) - 7 = -2 - 7 = -9
(ii) -x + 2 = -(-1) + 2 = 1 + 2 = 3
(iii) x² + 2x + 1 = (-1)² + 2(-1) + 1
= 1 - 2 + 1 = 0
(iv) 2x² - x - 2 = 2(-1)² - (-1) - 2
= 2(1) + 1 - 2 = 2 + 1 - 2 = 1
(i) a² + b² (ii) a² + ab + b² (iii) a² - b²
Substitute a = 2 and b = -2:
(i) a² + b² = (2)² + (-2)²
= 4 + 4 = 8
(ii) a² + ab + b² = (2)² + (2)(-2) + (-2)²
= 4 - 4 + 4 = 4
(iii) a² - b² = (2)² - (-2)²
= 4 - 4 = 0
(i) 2a + 2b (ii) 2a² + b² + 1 (iii) 2a²b + 2ab² + ab (iv) a² + ab + 2
Substitute a = 0 and b = -1:
(i) 2a + 2b = 2(0) + 2(-1) = 0 - 2 = -2
(ii) 2a² + b² + 1 = 2(0)² + (-1)² + 1
= 0 + 1 + 1 = 2
(iii) 2a²b + 2ab² + ab = 2(0)²(-1) + 2(0)(-1)² + (0)(-1)
= 0 + 0 + 0 = 0
(iv) a² + ab + 2 = (0)² + (0)(-1) + 2
= 0 + 0 + 2 = 2
(i) x + 7 + 4(x - 5)
(ii) 3(x + 2) + 5x - 7
(iii) 6x + 5(x - 2)
(iv) 4(2x - 1) + 3x + 11
(i) x + 7 + 4(x - 5)
= x + 7 + 4x - 20 = 5x - 13
Value at x = 2: 5(2) - 13 = 10 - 13 = -3
(ii) 3(x + 2) + 5x - 7
= 3x + 6 + 5x - 7 = 8x - 1
Value at x = 2: 8(2) - 1 = 16 - 1 = 15
(iii) 6x + 5(x - 2)
= 6x + 5x - 10 = 11x - 10
Value at x = 2: 11(2) - 10 = 22 - 10 = 12
(iv) 4(2x - 1) + 3x + 11
= 8x - 4 + 3x + 11 = 11x + 7
Value at x = 2: 11(2) + 7 = 22 + 7 = 29
(i) 3x - 5 - x + 9 (ii) 2 - 8x + 4x + 4 (iii) 3a + 5 - 8a + 1
(iv) 10 - 3b - 4 - 5b (v) 2a - 2b - 4 - 5 + a
(i) 3x - 5 - x + 9
= 2x + 4
Value at x = 3: 2(3) + 4 = 6 + 4 = 10
(ii) 2 - 8x + 4x + 4
= -4x + 6
Value at x = 3: -4(3) + 6 = -12 + 6 = -6
(iii) 3a + 5 - 8a + 1
= -5a + 6
Value at a = -1: -5(-1) + 6 = 5 + 6 = 11
(iv) 10 - 3b - 4 - 5b
= -8b + 6
Value at b = -2: -8(-2) + 6 = 16 + 6 = 22
(v) 2a - 2b - 4 - 5 + a
= 3a - 2b - 9
Value at a = -1, b = -2: 3(-1) - 2(-2) - 9 = -3 + 4 - 9 = -8
(ii) If p = -10, find the value of p² - 2p - 100.
(i) Simplification: z³ - 3z + 30
Substitute z = 10:
= (10)³ - 3(10) + 30
= 1000 - 30 + 30 = 1000
(ii) Expression: p² - 2p - 100
Substitute p = -10:
= (-10)² - 2(-10) - 100
= 100 + 20 - 100 = 20
Equation: 2x² + x - a = 5
Substitute x = 0:
2(0)² + 0 - a = 5
0 + 0 - a = 5
-a = 5 ⇒ a = -5
2(a² + ab) + 3 - ab
Simplify first:
= 2a² + 2ab + 3 - ab
= 2a² + ab + 3
Substitute a = 5 and b = -3:
= 2(5)² + (5)(-3) + 3
= 2(25) - 15 + 3
= 50 - 15 + 3 = 38
Exercise 12.4 (Number Patterns)
If the number of digits formed is taken to be n, the number of segments required to form n digits is given by the algebraic expression appearing on the right of each pattern. How many segments are required to form 5, 10, 100 digits of the kind 6, 4, 8?
The rules given in the NCERT book are:
For digit 6: Rule is 5n + 1
For digit 4: Rule is 3n + 1
For digit 8: Rule is 5n + 2
For Digit 6 (Pattern: 5n + 1):
n = 5: 5(5) + 1 = 26 segments
n = 10: 5(10) + 1 = 51 segments
n = 100: 5(100) + 1 = 501 segments
For Digit 4 (Pattern: 3n + 1):
n = 5: 3(5) + 1 = 16 segments
n = 10: 3(10) + 1 = 31 segments
n = 100: 3(100) + 1 = 301 segments
For Digit 8 (Pattern: 5n + 2):
n = 5: 5(5) + 2 = 27 segments
n = 10: 5(10) + 2 = 52 segments
n = 100: 5(100) + 2 = 502 segments
(See NCERT textbook for table format) Expressions are: 2n - 1, 3n + 2, 4n + 1, 7n + 20, n² + 1.
| Expression | 5th term | 10th term | 100th term |
|---|---|---|---|
| 2n - 1 | 2(5)-1 = 9 | 2(10)-1 = 19 | 2(100)-1 = 199 |
| 3n + 2 | 3(5)+2 = 17 | 3(10)+2 = 32 | 3(100)+2 = 302 |
| 4n + 1 | 4(5)+1 = 21 | 4(10)+1 = 41 | 4(100)+1 = 401 |
| 7n + 20 | 7(5)+20 = 55 | 7(10)+20 = 90 | 7(100)+20 = 720 |
| n² + 1 | 5²+1 = 26 | 10²+1 = 101 | 100²+1 = 10001 |
Chapter 12: Algebraic Expressions (Practice Questions)
RD Sharma / Extra Practice
(i) 3x - 7 (ii) -5x²y (iii) a² + b² - 2ab (iv) 4 + 5y²
(i) 3x - 7: Binomial (2 terms: 3x, -7)
(ii) -5x²y: Monomial (1 term)
(iii) a² + b² - 2ab: Trinomial (3 terms: a², b², -2ab)
(iv) 4 + 5y²: Binomial (2 terms: 4, 5y²)
Write expressions one below the other with like terms aligned:
7xy + 5yz - 3zx
+ 4yz + 9zx - 4y
+ -2xy - 3zx + 5x
---------------------------
5xy + 9yz + 3zx - 4y + 5x
= (7x² - 4xy + 8y² + 5x - 3y) - (5x² - 4y² + 6y - 3)
= 7x² - 4xy + 8y² + 5x - 3y - 5x² + 4y² - 6y + 3
= (7x² - 5x²) - 4xy + (8y² + 4y²) + 5x + (-3y - 6y) + 3
= 2x² - 4xy + 12y² + 5x - 9y + 3
Let the required expression be A.
(x² + xy + y²) + A = 2x² + 3xy
A = (2x² + 3xy) - (x² + xy + y²)
A = 2x² + 3xy - x² - xy - y²
A = x² + 2xy - y²
= 2x² - 2y² - 3x² - 3y² + 5x²
= (2x² - 3x² + 5x²) + (-2y² - 3y²)
= 4x² - 5y²
A + B = (3x² - 4x + 1) + (5x² + 3x - 8)
= 8x² - x - 7
Now, (A + B) - C = (8x² - x - 7) - (4x² - 7x + 3)
= 8x² - x - 7 - 4x² + 7x - 3
= 4x² + 6x - 10
Substitute x = -2:
= (-2)³ - (-2)² - (-2) - 1
= -8 - (4) + 2 - 1
= -8 - 4 + 2 - 1 = -12 + 1 = -11
We need to subtract the second expression from the first:
(93p² - 55p + 4) - (13p³ - 5p² + 17p - 90)
= 93p² - 55p + 4 - 13p³ + 5p² - 17p + 90
= -13p³ + (93p² + 5p²) + (-55p - 17p) + (4 + 90)
= -13p³ + 98p² - 72p + 94
Sum = (8a - 6a² + 9) + (-10a - 8 + 8a²)
= 2a² - 2a + 1
Now, subtract this from -3:
= -3 - (2a² - 2a + 1)
= -3 - 2a² + 2a - 1
= -2a² + 2a - 4
Solve inner brackets first:
= a - [b - {a - b + 1 + 3a}]
= a - [b - {4a - b + 1}]
= a - [b - 4a + b - 1]
= a - [-4a + 2b - 1]
= a + 4a - 2b + 1
= 5a - 2b + 1
Let sides be x, y, and z.
Perimeter = x + y + z ⇒ z = Perimeter - (x + y)
Sum of two sides (x + y) = (p² - 2p + 1) + (3p² - 5p + 3)
= 4p² - 7p + 4
Third side (z) = (6p² - 4p + 9) - (4p² - 7p + 4)
= 6p² - 4p + 9 - 4p² + 7p - 4
= 2p² + 3p + 5
= (x³ - x² + x + 1) - (-x³ + x² - x + 1)
= x³ - x² + x + 1 + x³ - x² + x - 1
= (x³ + x³) + (-x² - x²) + (x + x) + (1 - 1)
= 2x³ - 2x² + 2x
First simplify the expression:
= a² + ab + b² - a² + ab - b²
= (a² - a²) + (b² - b²) + (ab + ab)
= 2ab
Now substitute a=2, b=1:
= 2(2)(1) = 4
= 2x + 3y + 4x - 5y + 8z - z
= (2x + 4x) + (3y - 5y) + (8z - z)
= 6x - 2y + 7z
Determine the degree of each term:
4x³ → degree 3
-5x²y² → sum of powers = 2 + 2 = 4
7y → degree 1
2 → constant, degree 0
The highest degree among all terms is 4. Thus, the degree of the polynomial is 4.
Substitute a = 3, b = -2:
= (3)³ - (-2)³ - 3(3)(-2)(3 - (-2))
= 27 - (-8) - (-18)(3 + 2)
= 27 + 8 - (-18)(5)
= 35 - (-90) = 35 + 90 = 125
A regular hexagon has 6 equal sides.
Perimeter = 6 × side
= 6s
Shyam's age = x
3 times Shyam's age = 3x
Ram's age = 3 times Shyam's age + 5 = 3x + 5 years.
= (2a² - 7ab) - (a² - 3ab)
= 2a² - 7ab - a² + 3ab
= (2a² - a²) + (-7ab + 3ab)
= a² - 4ab
Start with the innermost bracket:
= 2x - [3y - {2x - y + x}]
= 2x - [3y - {3x - y}]
= 2x - [3y - 3x + y]
= 2x - [4y - 3x]
= 2x - 4y + 3x
= 5x - 4y
Chapter 12: Algebraic Expressions (Concepts & Summary)
1. Basic Definitions
- Variable: A letter (like x, y, z, a, b) used to represent an unknown quantity that can take any value.
- Constant: A symbol having a fixed numerical value (like -7, 5, 0, 100).
- Algebraic Expression: A mathematical phrase consisting of constants and
variables, combined along with fundamental mathematical operations like addition (+),
subtraction (-), multiplication (×) or division (÷). Example:
3x + 5
2. Terms, Factors, and Coefficients
- Terms: An expression is made of terms added or subtracted together. In 3x² - 5y, the terms are 3x² and -5y.
- Factors: Terms themselves are formed as a product of factors. Using a tree diagram, for the term 5xy, the factors are 5, x, y.
- Coefficient: The numerical factor of a term is called its numerical coefficient or simply coefficient. In -7xy, the coefficient of the term is -7.
3. Like and Unlike Terms
- Like Terms: Terms which have the exact same algebraic (variable) factors are
called like terms. Their numerical coefficients can be different.
Example: 5xy and -2xy are like terms. 3x²y and 7yx² are also like terms. - Unlike Terms: Terms which have different algebraic factors are called unlike
terms.
Example: 2x and 2y. 5x² and 5x.
4. Types of Algebraic Expressions
- Monomial: An expression which contains strictly one term.
Example: 5x, -7p²q. - Binomial: An expression which contains precisely two unlike terms.
Example: x + y, 2a - 5b. - Trinomial: An expression which contains exactly three unlike terms.
Example: x + y + z, a² + b² - 2ab. - Polynomial: In general, an expression with one or more unlike terms having non-negative integer exponents on the variables is called a polynomial. (So monomials, binomials, trinomials are all polynomials).
5. Addition and Subtraction
- Important Rule: Only Like terms can be combined (added or subtracted) to form a single term. Unlike terms cannot be combined into a single term.
- Addition: To add algebraic expressions, group the like terms together and sum
their numerical coefficients.
Example: (2x + 3y) + (4x + 5y) = (2 + 4)x + (3 + 5)y = 6x + 8y - Subtraction: To subtract an algebraic expression, change the sign of every term
in the expression being subtracted, and then add it.
Example: (5x - y) - (2x - 3y) = 5x - y - 2x + 3y = 3x + 2y
6. Finding the Value of an Expression
- The value of an algebraic expression depends purely on the values of the variables forming the expression.
- We find the value by simply replacing each variable with its given numerical value and applying
standard arithmetic operations.
Example: Find the value of x² + y² - xy when x = 2 and y = -1.
= (2)² + (-1)² - (2)(-1)
= 4 + 1 - (-2)
= 5 + 2 = 7