Question 1. Give five examples of expressions containing one variable and five examples of expressions containing two variables.
(a) Five examples containing one variable are:
(i) x – 4 (ii) 5 – y (iii) 2x + 5 (iv) 10x + 4 (v) 11 – z
(b) Five examples containing two variables are:
(i) 7x – 2y (ii) 5x + 2y – 10 (iii) 6x + z – 2 (iv) 15y – z (v) 9x – 10z
Question 2. Show on the number line x, x – 4, 2x + 1, 3x – 2.
Term, Factors and Coefficient
Consider the expression 3x + 7. This expression is containing two terms, i.e. 3x and 7.
Terms are added to form expressions.
Now consider the expression 7x2 – 5xy. This expression contains two terms, i.e. 7x2 and –5xy.
The term 7x2 is a product of 7, x and x. or we say that 7, x and x are factors of the term 7x2.
Also the term –5xy has the factors –5, x and y.
We know that a term is a product of its factors.
The numerical factor of term is called its coefficient. The expression 9xy – 7x has the two terms
9xy and –7x. The coefficient of the term 9xy is 9 and the coefficient of –7x is –7.
Question: Identify the coefficient of each term in the expression x2y2 – 10x2y + 5xy2 – 20.
Coefficient of x2y2 is 1.
Coefficient of x2y is –10.
Coefficient of xy2 is 5.
Monomials, Binomials and Polynomials
An expression containting only one term is called monomial.
Examples: 2x, 3y, 4p, q, 3x2, 4y2, 7xy, – 9m, –3, 4xyz, etc.
An expression containing two terms is called binomial.
Examples: a + b, x – y, 2p + 4q, 3xy – 7yz, p2 – 3q2, p2 – 3, a + 3, etc.
An expression containing three terms is called trinomial.
Examples: a + b + c, 4p + 9q – 3r, x2 + 4y2 – 3z2, etc.
An expression containing one or more terms with non-zero coefficient of variables having nonnegative exponents is called a polynomial.
Examples: a + b + c + d, 4xy, –5z, 5xyz – 10p, 4x + 3y – 3z, etc.
Question 1. Classify the following polynomials as monomials, binomials, trinomials –z + 5, x + y + z, y + z + 100, ab – ac, 17
–z + 5
ab – ac
x + y + z
y + z + 100
Question 2. Construct:
(a) 3 binomials with only x as a variable;
Sol: (a) 1. 5x – 4 2. 3x + 2 3. 10 + 2x
(b) 3 binomials with x and y as variables;
Sol: (b) 1. 2x + 3y 2. 5x – y 3. x – 3y
(c) 3 monomials with x and y as variables;
Sol: (c) 1. 2xy 2. xy 3. –7xy
(d) 2 polynomials with 4 or more terms.
Solution: (d) 1. 2x3 – x2 + 6x – 8 2. 10 + 7x – 2x2 + 4x3
Question: write two terms which are like
(i) 7xy (ii) 4mn2 (iii) 2l.
(i) Two terms like 7xy are: –3xy and 8xy.
(ii) Two terms like 4mn2 are: 6mn2 and –2n2m.
(iii) Two terms like 2l are: 5l and –7b.
Question 1. Identify the terms, their coefficients for each of the following expressions.
(i) 5xyz2 – 3zy (ii) 1 + x + x2 (iii) 4x2y2 – 4x2y2z2 + z2
(iv) 3 – pq + pr – rp (v) X/2 + X/2 - xy (vi) 0.3a – 0.6ab + 0.5b
(i) 5xyz2 – 3zy
(ii) 1 + x + x2
(iii) 4x2y2 – 4x2y2z2 + z2
(iv) 3 – pq + qr – rp
|Terms||3||–pq||+ qr||– rp|
(v) x/2 +x/2 - xy
(vi) 0.3a – 0.6ab + 0.5b
Question 2. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?
x + y, 1000, x + x2 + x3 + x4, 7 + y + 5x, 2y – 3y2, 2y – 3y2 + 4y3,
5x – 4y + 3xy, 4z – 15z2, ab + bc + cd + da, pqr, p2q + pq2, 2p + 2q
x + y
2y – 3y2
4z – 15z2
p2q + pq2
2p + 2q
7 + y + 5x
2y – 3y2 + 4y3
5x – 4y + 3xy
Following polynomials do not fit in any of these categories:
x + x2 + x3 + x2 [∵ It has 4 terms.]
ab + bc + cd + da [∵ It has 4 terms.]
Question 3. Add the following:
(i) ab – bc, bc – ca, ca – ab
(ii) a – b + ab, b – c + bc, c – a + ac
(iii) 2p2q2 – 3pq + 4, 5 + 7pq – 3p2q2
(iv) l2 + m2, m2 + n2, n2 + l2, 2lm + 2mn + 2nl
(a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3.
(b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz.
(c) Subtract 4p2q – 3pq + 5pq2 – 8p + 7q – 10
from 18 – 3p – 11q + 5pq – 2pq2 + 5p2q.
Solution: For subtraction we rearrange the terms such that the like terms are in the same column. Then we change the sign of the terms to be subtracted. Next we add the terms.
Product of Monomials
I. When we multiply monomials the product is also a monomial
II. We use the rules for exponents and powers when we multiply variables.
Example: Find the product of 2x, 5xy and 7x.
Solution: 2x * 5xy * 7z = (2 * 5 * 7) * (x * xy) * z
= 70 * (x2y 8 z)
= 70 * x2yz = 70x2yz
Question: Find 4x * 5y * 7z. First find 4x * 5y and multiply it by 7z; or first find 5y * 7z and multiply it by 4x. Is the result the same? What do you observe? Does the order in which you carry out the multiplication matter?
Solution: We have:
4x * 5y * 7z = (4x * 5y) * 7z
= 20xy * 7z = 140xyz
Also 4x * 5y * 7z = 4x * (5y 8 7z)
= 4x * 35yz = 140xyz
We observe that
(4x * 5y) * 7x = 4x(5y * 7z)
∴ The product of monomials is associative, i.e. the order in which we multiply the monomials does not matter.