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Chapter 6: Algebraic Expressions and Identities Exercise – 6.1

Question: 1

Identify the terms, their coefficients for each of the following expressions:

(i) 7x2yz − 5xy

(ii) x+ x + 1              

(iii) 3x2y− 5x2y2z+ z2

(iv) 9 – ab + bc – ca

(v) a2 + b2 − ab

(vi) 0.2x – 0.3xy + 0.5y

Solution:

Definitions:

A term in an algebraic expression can be a constant, a variable or a product of constants and variables separated by the signs of addition (+) or subtraction (-). Examples: 27, x, xyz, (1/2)x2yz etc.

The number factor of the term is called its coefficient.

(i) The expression 7x2yz − 5xy consists of two terms, i.e., 7x2yz and− 5xy.

The coefficient of 7x2yz is 7 and the coefficient of -5xy is - 5.

(ii) The expression x2 + x + 1 consists of three terms, i.e., x2, x and 1.

The coefficient of each term is 1.

(iii) The expression 3x2y2 – 5x2y2z2 + z2 consists of three terms, i.e., 3x2y2, – 5x2y2z2 and z2.

The coefficient of 3x2y2 is 3.

The coefficient of – 5x2y2z2 is -5 and the coefficient of z2 is 1.

(iv) The expression 9 – ab + bc – ca consists of four terms -9, - ab, bc and – ca.

The coefficient of the term 9 is 9.

The coefficient of -ab is -1.

The coefficient of bc is 1, and the coefficient of -ca is -1.

(v) The expression a2 + b2− ab consists of three terms , i.e., a2, b2 and− ab.

The coefficient of a2 is 12.

The coefficient of b2is12 and the coefficient of -ab is -1.

(vi) The expression 0.2x – 0.3xy + 0.5y consists of three terms, i.e., 0.2x, -0.3xy and 0.5y.

The coefficient of 0.2x is 0.2.

The coefficient of – 0.3xy is – 0.3, and the coefficient of 0.5y is 0.5.

 

Question: 2

Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category?

(i) x + y

(ii) 1000

(iii) x + x2 + x3 + x4

(iv) 7 + a + 5b

(v) 2b −3b2

(vi) 2y − 3y+ 4y3

(vii) 5x – 4y + 3x

(viii) 4a − 15a2

(ix) xy + yz + zt + tx

(x) pqr

(xi) p2q + pq2

(xii) 2p + 2q

Solution:

Definitions:

A polynomial is monomial if it has exactly one term. It is called binomial if it has exactly two non-zero terms. A polynomial is a trinomial if it has exactly three non-zero terms.

(i) The polynomial x + y has exactly two non-zero terms, i.e., x and y. Therefore, it is a binomial.

(ii) The polynomial 1000 has exactly one term, i.e., 1000. Therefore, it is a monomial.

(iii) The polynomial x + x2 + x3 + x4 has exactly four terms, i.e., x, x2, x3 and x4. Therefore, it doesn’t belong to any of the categories.

(iv) The polynomial 7 + a + 5b has exactly three terms, i.e., 7, a and 5b. Therefore, it is a trinomial.

(v) The polynomial 2b – 3b2 has exactly two terms, i.e., 2b and -3b2. Therefore, it is a binomial.

(vi) The polynomial 2y–3y2+4y3 has exactly three terms, i.e., 2y, 3y2 and 4y3.. Therefore, it is a trinomial.

(vii) The polynomial 5x – 4y + 3x has exactly three terms, i.e., 5x, – 4y and 3x. Therefore, it is a trinomial.

(viii) The polynomial 4a – 15a2 has exactly two terms, i.e., 4a and -15a2. Therefore, it is a binomial.

(ix) The polynomial xy + yz + zt + tx has exactly four terms xy, yz, zt and tx. Therefore, it doesn’t belong to any of the categories.

(x) The polynomial pqr has exactly one term, i.e., pqr. Therefore, it is a monomial.

(xi) The polynomial p2q + pq2 has exactly two terms, i.e., p2q and pq2. Therefore, it is a binomial.

(xii) The polynomial 2p+ 2q has two terms, i.e., 2p and 2q. Therefore, it is a binomial.


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