Exercise 12.1 Page: 234

**1. Get the algebraic expressions in the following cases using variables, constants and arithmetic operations.**

**(i) Subtraction of z from y.**

**Solution:-**

= Y – z

**(ii) One-half of the sum of numbers x and y.**

**Solution:-**

= ½ (x + y)

= (x + y)/2

**(iii) The number z multiplied by itself.**

**Solution:-**

= z × z

= z^{2}

**(iv) One-fourth of the product of numbers p and q.**

**Solution:-**

= ¼ (p × q)

= pq/4

**(v) Numbers x and y both squared and added.**

**Solution:-**

= x^{2 }+ y^{2}

**(vi) Number 5 added to three times the product of numbers m and n.**

**Solution:-**

= 3mn + 5

**(vii) Product of numbers y and z subtracted from 10.**

**Solution:-**

= 10 – (y × z)

= 10 – yz

**(viii) Sum of numbers a and b subtracted from their product.**

**Solution:-**

= (a × b) – (a + b)

= ab – (a + b)

**2. (i) Identify the terms and their factors in the following expressions**

**Show the terms and factors by tree diagrams.**

**(a) x – 3**

**Solution:-**

Expression: x – 3

Terms: x, -3

Factors: x; -3

**b) 1 + x + x ^{2}**

**Solution:-**

Expression: 1 + x + x^{2}

Terms: 1, x, x^{2}

Factors: 1; x; x,x

**(c) y – y ^{3}**

**Solution:-**

Expression: y – y^{3}

Terms: y, -y^{3}

Factors: y; -y, -y, -y

**(d) 5xy ^{2} + 7x^{2}y**

**Solution:-**

Expression: 5xy^{2} + 7x^{2}y

Terms: 5xy^{2}, 7x^{2}y

Factors: 5, x, y, y; 7, x, x, y

**(e) – ab + 2b ^{2} – 3a^{2}**

**Solution:-**

Expression: -ab + 2b^{2} – 3a^{2}

Terms: -ab, 2b^{2}, -3a^{2}

Factors: -a, b; 2, b, b; -3, a, a

**(ii) Identify terms and factors in the expressions given below:**

**(a) – 4x + 5 (b) – 4x + 5y (c) 5y + 3y ^{2} (d) xy + 2x^{2}y^{2}**

**(e) pq + q (f) 1.2 ab – 2.4 b + 3.6 a (g) ¾ x + ¼**

**(h) 0.1 p ^{2} + 0.2 q^{2}**

**Solution:-**

Expressions is defined as, numbers, symbols and operators (such as +. – , × and ÷) grouped together that show the value of something.

In algebra a term is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or – signs or sometimes by division.

Factors is defined as, numbers we can multiply together to get another number.

Sl.No. |
Expression |
Terms |
Factors |

(a) |
– 4x + 5 | -4x
5 |
-4, x
5 |

(b) |
– 4x + 5y | -4x
5y |
-4, x
5, y |

(c) |
5y + 3y^{2} |
5y
3y |
5, y
3, y, y |

(d) |
xy + 2x^{2}y^{2} |
xy
2x |
x, y
2, x, x, y, y |

(e) |
pq + q | pq
q |
P, q
Q |

(f) |
1.2 ab – 2.4 b + 3.6 a | 1.2ab
-2.4b 3.6a |
1.2, a, b
-2.4, b 3.6, a |

(g) |
¾ x + ¼ | ¾ x
¼ |
¾, x
¼ |

(h) |
0.1 p^{2} + 0.2 q^{2} |
0.1p^{2}
0.2q |
0.1, p, p
0.2, q, q |

**3. Identify the numerical coefficients of terms (other than constants) in the following expressions:**

**(i) 5 – 3t ^{2} (ii) 1 + t + t^{2} + t^{3} (iii) x + 2xy + 3y (iv) 100m + 1000n (v) – p^{2}q^{2} + 7pq (vi) 1.2 a + 0.8 b (vii) 3.14 r^{2} (viii) 2 (l + b)**

**(ix) 0.1 y + 0.01 y ^{2}**

**Solution:-**

Expressions is defined as, numbers, symbols and operators (such as +. – , × and ÷) grouped together that show the value of something.

In algebra a term is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or – signs or sometimes by division.

A coefficient is a number used to multiply a variable (2x means 2 times x, so 2 is a coefficient) Variables on their own (without a number next to them) actually have a coefficient of 1 (x is really 1x)

Sl.No. |
Expression |
Terms |
Coefficients |

(i) |
5 – 3t^{2} |
– 3t^{2} |
-3 |

(ii) |
1 + t + t^{2} + t^{3} |
t
t t |
1
1 1 |

(iii) |
x + 2xy + 3y | x
2xy 3y |
1
2 3 |

(iv) |
100m + 1000n | 100m
1000n |
100
1000 |

(v) |
– p^{2}q^{2} + 7pq |
-p^{2}q^{2}
7pq |
-1
7 |

(vi) |
1.2 a + 0.8 b | 1.2a
0.8b |
1.2
0.8 |

(vii) |
3.14 r^{2} |
3.14^{2} |
3.14 |

(viii) |
2 (l + b) | 2l
2b |
2
2 |

(ix) |
0.1 y + 0.01 y^{2} |
0.1y
0.01y |
0.1
0.01 |

**4. (a) Identify terms which contain x and give the coefficient of x.**

**(i) y ^{2}x + y (ii) 13y^{2} – 8yx (iii) x + y + 2**

**(iv) 5 + z + zx (v) 1 + x + xy (vi) 12xy ^{2} + 25**

**(vii) 7x + xy ^{2}**

**Solution:-**

Sl.No. |
Expression |
Terms |
Coefficient of x |

(i) |
y^{2}x + y |
y^{2}x |
y^{2} |

(ii) |
13y^{2} – 8yx |
– 8yx | -8y |

(iii) |
x + y + 2 | x | 1 |

(iv) |
5 + z + zx | x
zx |
1
z |

(v) |
1 + x + xy | xy | y |

(vi) |
12xy^{2} + 25 |
12xy^{2} |
12y^{2} |

(vii) |
7x + xy^{2} |
7x
xy |
7
y |

**(b) Identify terms which contain y ^{2} and give the coefficient of y^{2}.**

**(i) 8 – xy ^{2} (ii) 5y^{2} + 7x (iii) 2x^{2}y – 15xy^{2} + 7y^{2}**

**Solution:-**

Sl.No. |
Expression |
Terms |
Coefficient of y^{2} |

(i) |
8 – xy^{2} |
– xy^{2} |
– x |

(ii) |
5y^{2} + 7x |
5y^{2} |
5 |

(iii) |
2x^{2}y – 15xy^{2} + 7y^{2} |
– 15xy^{2}
7y |
– 15x
7 |

**5. Classify into monomials, binomials and trinomials.**

**(i) 4y – 7z**

**Solution:-**

Binomial.

An expression which contains two unlike terms is called a binomial.

**(ii) y ^{2}**

**Solution:-**

Monomial.

An expression with only one term is called a monomial.

**(iii) x + y – xy**

**Solution:-**

Trinomial.

An expression which contains three terms is called a trinomial.

**(iv) 100**

**Solution:-**

Monomial.

An expression with only one term is called a monomial.

**(v) ab – a – b**

**Solution:-**

Trinomial.

An expression which contains three terms is called a trinomial.

**(vi) 5 – 3t**

**Solution:-**

Binomial.

An expression which contains two unlike terms is called a binomial.

**(vii) 4p ^{2}q – 4pq^{2}**

**Solution:-**

Binomial.

An expression which contains two unlike terms is called a binomial.

**(viii) 7mn**

**Solution:-**

Monomial.

An expression with only one term is called a monomial.

**(ix) z ^{2} – 3z + 8**

**Solution:-**

Trinomial.

An expression which contains three terms is called a trinomial.

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

**Solution:-**

Binomial.

An expression which contains two unlike terms is called a binomial.

**(xi) z ^{2} + z**

**Solution:-**

Binomial.

An expression which contains two unlike terms is called a binomial.

**(xii) 1 + x + x ^{2}**

**Solution:-**

Trinomial.

An expression which contains three terms is called a trinomial.

**6. State whether a given pair of terms is of like or unlike terms.**

**(i) 1, 100**

**Solution:-**

Like term.

When term have the same algebraic factors, they are like terms.

**(ii) –7x, (5/2)x**

**Solution:-**

Like term.

When term have the same algebraic factors, they are like terms.

**(iii) – 29x, – 29y**

**Solution:-**

Unlike terms.

The terms have different algebraic factors, they are unlike terms.

**(iv) 14xy, 42yx**

**Solution:-**

Like term.

When term have the same algebraic factors, they are like terms.

**(v) 4m ^{2}p, 4mp^{2}**

**Solution:-**

Unlike terms.

The terms have different algebraic factors, they are unlike terms.

**(vi) 12xz, 12x ^{2}z^{2}**

**Solution:-**

Unlike terms.

The terms have different algebraic factors, they are unlike terms.

**7. Identify like terms in the following:**

**(a) – xy ^{2}, – 4yx^{2}, 8x^{2}, 2xy^{2}, 7y, – 11x^{2}, – 100x, – 11yx, 20x^{2}y, – 6x^{2}, y, 2xy, 3x**

**Solution:-**

When term have the same algebraic factors, they are like terms.

They are,

– xy^{2}, 2xy^{2}

– 4yx^{2}, 20x^{2}y

8x^{2}, – 11x^{2}, – 6x^{2}

7y, y

– 100x, 3x

– 11yx, 2xy

**(b) 10pq, 7p, 8q, – p ^{2}q^{2}, – 7qp, – 100q, – 23, 12q^{2}p^{2}, – 5p^{2}, 41, 2405p, 78qp,**

**13p ^{2}q, qp^{2}, 701p^{2}**

**Solution:-**

When term have the same algebraic factors, they are like terms.

They are,

10pq, – 7qp, 78qp

7p, 2405p

8q, – 100q

– p^{2}q^{2}, 12q^{2}p^{2}

– 23, 41

– 5p^{2}, 701p^{2}

13p^{2}q, qp^{2}