Q1: Identify the terms, their coefficients for each of the following expressions:

(i) \(7x^{2}yz-5xy\)

(ii) \(x^{2}+x+1\)

(iii) \(3x^{2}y^{2}-5x^{2}y^{2}z^{2}+z^{2}\)

(iv) 9 â€“ ab + bc â€“ ca

(v) \(\frac{a}{2}+\frac{b}{2}-ab\)

(vi) 0.2x â€“ 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, \(\frac{1}{2}x^{2} y z\)

The number factor of the term is called its coefficient.

(i) The expression \(7x^{2}y z-5x y\)

The coefficient of \(7x^{2} y z\)

(ii) The expression x^{2} + x + 1 consists of three terms, i.e., x^{2}, x and 1.

The coefficient of each term is 1.

(iii) The expression 3x^{2}y^{2} â€“ 5x^{2}y^{2}z^{2} + z^{2} consists of three terms, i.e., 3x^{2}y^{2}, â€“ 5x^{2}y^{2}z^{2} and z^{2}.

The coefficient of 3x^{2}y^{2} is 3.

The coefficient of â€“ 5x^{2}y^{2}z^{2} is -5 and the coefficient of z^{2} is 1.

(iv) The expression 9 â€“ ab + bc â€“ 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 \(\frac{a}{2}+\frac{b}{2}-ab\)

The coefficient of Â \(\frac{a}{2}\; is\; \frac{1}{2}\)

The coefficient of \(\frac{b}{2}\; is\; \frac{1}{2}\)

(vi) The expression 0.2x â€“ 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.

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Q2) Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category?

(i) x + y

(ii) 1000

(iii) \(x+x^{2}+x^{3}+x^{4}\)

(iv) 7 + a + 5b

(v) \(2b-3b^{2}\)

(vi) \(2y-3y^{2}+4y^{3}\)

(vii) 5x â€“ 4y + 3x

(viii) \(4a-15a^{2}\)

(ix) xy + yz + zt + tx

(x) pqr

(xi) \(p^{2}q+pq^{2}\)

(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 \(x+x^{2}+x^{3}+x^{4}\)^{2}, \(x^{3}\; and\; x^{4}\)

(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 â€“ 3b^{2} has exactly two terms, i.e., 2b and -3b^{2}. Therefore, it is a binomial.

(vi) The polynomial \(2y – 3y^{2} + 4y^{3}\)^{2} and \(4y^{3}\)

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

(viii) The polynomial 4a â€“ 15a^{2} has exactly two terms, i.e., 4a and -15a^{2}. 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 p^{2}q + pq^{2} has exactly two terms, i.e., p^{2}q and pq^{2}. Therefore, it is a binomial.

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