Question

If a,B are the Zeroes of a polynomial, such that a+B=6 and aB=4 , then Write the polynomial ​

Answer 1

Answer :- x² - 6x + 4.

Concept

• If S is the sum of roots of a quadratic polynomial and P is its product, the quadratic polynomial is given by x² - Sx + P.

Here, S = 6 and P = 4.

Therefore, quadratic polynomial = x² - Sx + P

= x² - 6x + 4

Additional details

• The choice of variable is immaterial. I have chosen x, likewise, once could take it as y , t or anything else.

• Therefore, if we write the polynomial as y² - 6y + 4 , it is exactly same as x² - 6x + 4.

Answer 2

We know,

• If S represents the Sum of zeroes

and

• P represents the Product of zeroes,

then required quadratic polynomial is given by

$\bigstar \: \: \boxed{ \pink{ \rm \: f(x) = k \: ( {x}^{2} \: - \: Sx \: + \: P) \: \: where \: k \: \ne \: 0 \: }}$

According to statement,

• a and B are the zeroes of a polynomial such that

$\begin{gathered}\begin{gathered}\bf \: \begin{cases} &\sf{a \: + \: B \: = \: S \: = 6} \\ &\sf{aB \: = \: P \: = \: 4} \end{cases}\end{gathered}\end{gathered}$

So,

• The required polynomial is given by

$\rm : \implies \:f(x) = k \: ( {x}^{2} \: - \: Sx \: + \: P) \: \: where \: k \: \ne \: 0 \:$

$\bigstar \: \: \boxed{ \pink{ \rm \: f(x) = k \: ( {x}^{2} \: - \: 6x \: + \: 4) \: \: where \: k \: \ne \: 0 \:}}$

Additional Information

• In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. 

• The word polynomial is derived from the Greek words ‘poly’ means ‘many‘ and ‘nominal’ means ‘terms‘, so altogether it said “many terms”. A polynomial can have any number of terms but not infinite.

• The degree of a polynomial is defined as the highest degree of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.

• Monomial :- A monomial is an expression which contains only one term. For an expression to be a monomial, the single term should be a non-zero term.

• Binomial :- A binomial is a polynomial expression which contains exactly two terms. A binomial can be considered as a sum or difference between two monomials. 

• Trinomial :- A trinomial is a polynomial expression which contains exactly three terms.