Page 139 - ICSE Math 8

To find the degree of a polynomial in two or more variables, find the sum of Try This
the powers of the variables in each term. The highest sum is the degree of
the polynomial. For example, Find the degree of the following
3
(a) 5x y + 5xy – 20 is a polynomial in x and y. polynomials. 3
3
Sum of the powers of x and y in 5x y = 3 + 1 = 4 (a) –2x + 5x – 8
5
4 2
(b) –4x + 5x y
Sum of the powers of x and y in 5xy = 1 + 1 = 2 (c) –5x + 9
Sum of the powers of x and y in –20 = 0 + 0 = 0 (d) –125
As the highest sum is 4, therefore the degree of the given polynomial is 4.
3 3
5 4
(b) 13a b + 19a b + 7 is a polynomial in a and b with degree 9.
There are different types of polynomials according to their degrees.
(a) A polynomial of degree 1 is called a linear polynomial. For example, p + 9 is a linear polynomial in
one variable (p) and –2x + 3y is a linear polynomial in two variables (x and y).
2
(b) A polynomial of degree 2 is called a quadratic polynomial. For example, x + 3x – 9 is a quadratic polynomial
2
in one variable (x) and 3a + 5ab – bc is a quadratic polynomial in three variables (a, b and c).
2
3
(c) A polynomial of degree 3 is called a cubic polynomial. For example, –8a + a is a cubic polynomial
2
2
2
in one variable (a) and 4x – 3xy + y x is a cubic polynomial in two variables (x and y).
–3
4
3
(d) A polynomial of degree 4 is called a quartic polynomial. For example, 5 a + 8a – 9 is a quartic
3
3
3
polynomial in one variable (a) and 5a + 5a b + b c + abcd is a quartic polynomial in four variables
(a, b, c and d).
Example 1: Classify the given expressions as monomial, binomial or trinomial.
2
3
(a) 3x y + 7 (b) 9ab + yx + 7x (c) 8
2
Solution: (a) The given expression is a binomial since the number of terms is 2 (3x y and 7).
(b) The given expression is a trinomial since the number of terms is 3
(9ab, yx and 7x).
(c) The given expression is a monomial since the number of term is 1 (8).
2
Example 2: Find the coefficient of ab in each of the following terms. Try This
–2
2
(a) 5ab (b) 17xab 2 (c) 9 ab 2 Which of these are not polynomials?
2
2
Solution: (a) Coefficient of ab in 5ab = 5 (a) a – 5a – + a 3
2
1 2
2
2
(b) Coefficient of ab in 17xab = 17x 2 2 9
2
–2 –2 (b) 5x – 9x (c) 45 (d) x + 1
2
2
(c) Coefficient of ab in ab = 1 + x x 2
9 9
EXERCISE 12.1
1. Separate the constants and variables from the following:
2
2
7a, bc , –73, 16b , 49, xyz
2. Write the number of terms in each of the following polynomials.
2
a + 6a – 7
(a) 5 3 2 (b) 6xy + 17 x – 7y + 3 (c) 21 (d) –7xy – 3y 2
7 19 11
3. Separate the monomials, binomials and trinomials from the following expressions.
3
3 2
3
3
2
3
3
4
9 10
7x y , 82 + yz – z , x + 3a b, 4x ÷ 5, x – y, y – 7z y + x , 19x y × 7xyz – yz ÷ 4
6
2 3
4. In 13 ab c , find the coefficient of
3
2 3
(a) 6 a (b) 6a (c) ab 2 (d) b c (e) 6 c (f) 6ab 2
13 13
127