Find out the degree of the polynomials and the leading coefficients of the polynomials given below:

(i)

(ii) 13x³ – x¹³ – 113

(iii) -77 + 7x² – x⁷

(iv) -181 + 0.8y – 8y² + 115y³ + y⁸

(v) x⁷ – 2x³y⁵ + 3xy⁴ – 10xy + 10

(i) The monomials in the polynomial are called the terms. The highest power of the terms is the degree of the polynomial.

x² – 2x³ + 5x⁷ – x³ – 70x – 8 is a polynomial in x. Here we have 6 monomials x², – 2x³, + 5x⁷, –x³, –70x and –8 which are called the terms of the polynomial.

The highest power is 7 so the degree of the polynomial is 7.

(ii) 13x³ – x¹³ – 113 is a polynomial in x. Here we have 3 monomials and the highest power is 13 so the degree of the polynomial is 13.

(iii) -77 + 7x² – x⁷ is a polynomial in x. Here we have 3 monomials and the highest power is 7 so the degree of the polynomial is 7.

(iv) -181 + 0.8y – 8y² + 115y³ + y⁸ is a polynomial in x. Here we have 5 monomials and the highest power is 8 so the degree of the polynomial is 8.

(v) x⁷ – 2x³y⁵ + 3xy⁴ – 10xy + 10 is a polynomial in x and y, Here we have 5 monomials.

Term 1: x⁷ variable x, power of x is 7. Hence the power of the term is 7.

Term 2: – 2x³y⁵ the variables are x and y; the power of x is 3 and the power of y is 5.

Hence the power of the term – 2x³y⁵ is 3 + 5 = 8 [Sum of the exponents of variables x and y ].

Term 3: 3xy⁴ the variables are x and y; the power of x is 1 and the power of y is 4.

Hence the power of the term 3xy⁴ is 1 + 4 = 5 [Sum of the exponents of variables x and y].

Term 4: – 10xy the variables are x and y; the power of x is 1 and the power of y is 1.

Hence the power of the term -10xy is 1 + 1 = 2 [Sum of the exponents of variables x and y].

Term 5: 10 the constant term and it can be written as 10x⁰y⁰. The power of the variables x⁰y⁰ is zero. Hence the power of the term 10 is 0.

So the highest power is 8, hence the degree of the polynomial is 8.