Question

If a+b=1, prove that a^3+b^3+3ab=1.

Answer 1

Answer- The above question is from the chapter 'Polynomials'.

Let's know about polynomials first.

Polynomial- It is an algebraic expression involving use of variables and constants.

p(x)- It is used to denote a polynomial. It is read is 'Polynomial in x'.

Polynomials can be classified on two basis:

1) Number of terms

E.g.- Polynomial with one term is called monomial.

Two terms- binomial

Three terms- trinomial

2) Power of variable

E.g.- Polynomial with degree 1 is called linear polynomial.

degree 2- quadratic polynomial

degree 3- cubic polynomial

degree 4- biquadratic polynomial

Concept used: (x + y)³ = x³ + y³ + 3xy(x + y)

Given question: If a + b = 1, prove that a³ +b³ + 3ab = 1.

Solution: a + b = 1 (Given)

Cubing both sides, we get,

(a + b)³ = 1³

a³ + b³ + 3ab(a + b) = 1 ∵ (x + y)³ = x³ + y³ + 3xy(x + y) where x = a and y = b

a³ + b³ + 3ab × 1 = 1     ∵ a + b = 1 (Given)

⇒ a³ + b³ + 3ab = 1

Hence, proved.