# Algebraic Expressions

## Exercise 12.1

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

(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.

Solution:

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

Ans:* y* – *z*

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

Ans: ½(*x* + *y*)

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

Ans: *z* X *z*

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

Ans: ¼ *pq*

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

*Ans: x*^{2} + *y*^{2}

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

Ans: 5 + 3*mn*

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

Ans: 10 – *yz*

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

Ans: *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 (b) 1 + *x *+ *x*^{2} (c) *y *– *y*^{3}

(d) 5*xy*^{2} + 7*x*^{2}*y *(e) – *ab *+ 2*b*^{2} – 3*a*^{2}

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

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

(d) *xy *+ 2*x*^{2}*y*^{2} (e) *pq *+ *q *(f) 1.2*ab *– 2.4*b *+ 3.6*a*

(g)3/4*x *+1/4 (h) 0.1 *p*^{2} + 0.2 *q*^{2}

Solution:

(a) – 4*x *+ 5

Terms: – 4*x*, 5

Factors: (– 4, *x*) (5)

(b) – 4*x *+ 5*y *

Terms: – 4*x*, 5*y*

Factors: (– 4, *x*) (5, *y*)

* *(c) 5*y *+ 3*y*^{2}

Terms: 5*y *, 3*y*^{2}

Factors: (5, *y*), (3, *y, y *)

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

Terms: *xy *, 2*x*^{2}*y*^{2}

Factors: (*x,y*)* *(2, *x, x, y, y*)

(e) *pq *+ *q *

Terms: *pq *, *q*

Factors: (*p, q*) ( *q *)

(f) 1.2*ab *– 2.4*b *+ 3.6*a*

Terms: 1.2*ab*, * *– 2.4*b *, 3.6*a*

Factors: (1.2, *a, b*) ( – 2.4, *b *) (3.6, *a*)

(g) ^{3}/_{4 }*x *+ ^{1}/_{4}

Terms: ^{3}/_{4 }*x *, ^{1}/_{4}

Factors: (^{3}/_{4 , }*x *) (^{1}/_{4)}

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

Terms: 0.1*p*^{2}, 0.2*q*^{2}

Factors: (0.1, *p, p *)* *( 0.2, *q, q*)

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

(i) 5 – 3*t*^{2} (ii) 1 + *t *+ *t*^{2} + *t*^{3} (iii) *x *+ 2*xy *+ 3*y*

(iv) 100*m *+ 1000*n *(v) – *p*^{2}*q*^{2} + 7*pq *(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:

(i) 5 – 3*t*^{2}

Term: – 3*t*^{2}

The coefficient of – 3*t*^{2 } is – 3

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

Terms: *t *, *t*^{2} , *t*^{3}

The coefficient of *t *is 1

The coefficient of *t*^{2} is 1

The coefficient of *t*^{3} is 1

(iii) *x *+ 2*xy *+ 3*y*

Terms: *x *, 2*xy *, 3*y*

The coefficient of *x *is 1

The coefficient of 2*xy* is 2

The coefficient of 3*y *is 3

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

Terms: 100*m *, 1000*n*

The coefficient of 100*m *is 100

The coefficient of 1000*n* is 1000

* *(v) – *p*^{2}*q*^{2} + 7*pq *

Terms: – *p*^{2}*q*^{2} , 7*pq*

The coefficient of – *p*^{2}*q*^{2} is – 1

The coefficient of 7*pq* is 7

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

Terms: 1.2*a *, 0.8*b*

The coefficient of 1.2*a x *is 1.2

The coefficient of 0.8*b* is 0.8

(vii) 3.14 *r*^{2}

Terms: 3.14 *r*^{2}

The coefficient of 3.14 *r*^{2} is 3.14

(viii) 2 (*l *+ *b*)

2 (*l *+ *b*) = 2*l *+ 2*b*

Terms: 2*l *, 2*b*

The coefficient of 2*l *is 2

The coefficient of 2*b* is 2

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

Terms: 0.1*y *, 0.01*y*^{2}

The coefficient of 0.1*y *is 0.1

The coefficient of 0.01*y*^{2}is 0.01

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

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

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

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

Solution:

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

Terms containing *x*: *y*^{2}*x*

coefficient of *x* is *y*^{2}

(ii) 13*y*^{2} – 8*yx *

Terms containing *x*: – 8*yx *

coefficient of *x* is – 8*y*

(iii) *x *+ *y *+ 2

Terms containing *x*: *x *

coefficient of *x* is 1

(iv) 5 + *z *+ *zx *

* *Terms containing *x*: *zx *

coefficient of *x* is *z*

* *(v) 1 + *x *+ *xy *

Terms containing *x*: *x*, *xy*

coefficient of *x* is 1

coefficient of *x *is *y*

(vi) 12*xy*^{2} + 25

Terms containing *x*: 12*xy*^{2}

coefficient of *x* is 12*y*^{2}

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

Terms containing *x*: 7*x*, *xy*^{2}

coefficient of *x* is 7

coefficient of *x *is *y*^{2}

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

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

Solution:

(i) 8 – *xy*^{2}

Terms containing *y*^{2}: – *xy*^{2}

coefficient of *y*^{2} is – *x*

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

Terms containing *y*^{2}: 5*y*^{2}

coefficient of *y*^{2} is 5

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

Terms containing *y*^{2}: – 15*xy*^{2}, 7*y*^{2}

coefficient of *y*^{2} is – 15*x*

coefficient of *y*^{2} is 7

5). Classify into monomials, binomials and trinomials.

(i) 4*y *– 7*z *(ii) *y*^{2} (iii) *x *+ *y *– *xy *

(iv) 100 (v) *ab *– *a *– *b *(vi) 5 – 3*t *

(vii) 4*p*^{2}*q *– 4*pq*^{2} (viii) 7*mn *

(ix) *z*^{2} – 3*z *+ 8 (x) *a*^{2} + *b*2 (xi) *z*^{2} + *z *(xii) 1 + *x *+ *x*^{2}

Solution:

(i) 4*y *– 7*z *

Ans: It is binomial

(ii) *y*^{2}

Ans: It is monomial

(iii) *x *+ *y *– *xy *

Ans: It is trinomial

(iv) 100

Ans: It is monomial

(v) *ab *– *a *– *b *

Ans: It is trinomial

(vi) 5 – 3*t *

Ans: It is binomial

(vii) 4*p*^{2}*q *– 4*pq*^{2}

Ans: It is binomial

(viii) 7*mn *

Ans: It is monomial

(ix) *z*^{2} – 3*z *+ 8

Ans: It is trinomial

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

Ans: It is binomial

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

Ans: It is binomial

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

Ans: It is trinomial

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

(i) 1, 100 (ii) –7*x*, 5/2 *x *(iii) – 29*x*, – 29*y*

(iv) 14*xy*, 42*yx *(v) 4*m*^{2}*p*, 4*mp*^{2} (vi) 12*xz*, 12*x*^{2}*z*^{2}

Solution:

(i) 1, 100

Ans: They are like terms

(ii) –7*x*, 5/2 *x *

Ans: They are like terms

(iii) – 29*x*, – 29*y*

Ans: They are unlike terms

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

Ans: They are like terms

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

Ans: They are unlike terms

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

Ans: They are unlike terms

7). Identify like terms in the following:

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

– 6*x*^{2}, *y*, 2*xy*, 3*x*

Ans: (– *xy*^{2}, 2*xy*^{2} ); ( – 4*yx*^{2}, 20*x*^{2}*y* ); ( 8*x*^{2}, – 11*x*^{2}, – 6*x*^{2}); ( 7*y*, *y* );

( – 100*x*, 3*x* ); ( – 11*yx*, 2*xy* )

(b) 10*pq*, 7*p*, 8*q*, – *p*^{2}*q*^{2}, – 7*qp*, – 100*q*, – 23, 12*q*^{2}*p*^{2}, – 5*p*^{2}, 41, 2405*p*, 78*qp*, 13*p*^{2}*q*, *qp*^{2}, 701*p*^{2}

Ans: (10*pq*, – 7*qp*, 78*qp* ); (7*p*, 2405*p*); (8*q*, – 100*q* ); (– 23, 41);

( – *p*^{2}*q*^{2}, 12*q*^{2}*p*^{2} ); ( – 5*p*^{2}, 701*p*^{2}); ( 13*p*^{2}*q*, *qp*^{2} )

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