How many solution of equation x^2+y^2=1?
Q.1.How many solution of equation x^2+y^2=1?
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Special question of sum and product of roots
v v Q. (1). If 1,2,3 are the roots of the equation x 3 +ax 2 +bx+c=0 then what is the value of c? Ans. Apply product of roots αβγ= -c/a=- constant term/coefficient of x 3 1x2x3 = -c/1 → 6 = -c → c= - 6 Q.2. If (x+√2) is a factor of kx 2 -√2 x + 1, then what is the value of k? Ans. Putting value of x in given equation so x+ √ 2=0 then x= - √ 2 → k (- √ 2) 2 - √ 2 × (-√ 2) +1 = 0 2k+2+1=0 2k+3=0 → 2k=-3 → k = -3/2 Q.3. What is the factor of the equation (x² -5x) ² -30 (x² -5x) -216=0? Ans. First of all, let x 2 –5x =y then equation become y 2 –30y-216=0 then factorize 216 in two term for example 216=36×6 so, can be written 30y = 36y-6y so y 2 -(36y-6y)-2016=0 y 2 -36y+6y-216=0 taking common we get y(y-36) +6(y-36) =0 so (y-36) (y-6) =0 then y has two values
Sum and product of roots/zeroes
Sum and Product of roots/zeroes of a polynomial- Standard quadratic equation of a polynomial is ax 2 +bx+c=0, where a, b,c is constant. Let α and β are roots of a polynomial then factors are (x- α ) (x- β) = 0 x 2 -xβ-xα+αβ=0 x 2 -x(α+β) +αβ=0 Compare standard equation of polynomial then α+β= - b/a = - coefficient of x/coefficient of x 2 αβ = c/a = constant term/coefficient of x 2 Q.1. The roots of the equation x²-2x+3=0 are α and β. What is the equation whose roots are α², β²? Ans. Let α and β are the roots of the equation x²-2x+3=0 then sum and product of the roots α+β= - b/a = - coefficient of x/coefficient of x²=-(-2)/1=2 αβ = c/a = constant term/coefficient of x² =3/1=3 then
Introduction of polynomial
Polynomial – An expression containing variable and constant, power of variable must be whole number. For example 3+x 2 , 4+6x 2 Standard representation of polynomial-- a 0 +a 1 x+a 2 x 2 +a 3 x 3 +-----------a n x n where a 0 , a 1 , a 2 , ……… a n are real number and also called coefficient of polynomial and n is whole number. Checking polynomial----- its true or false ü X 2 +2x+1 ü X+1/x √ x+1/2 Degree of polynomial— In polynomial highest power of variable is called degree of polynomial. Based on degree there are mainly three types of polynomial 1 .Linear polynomial — a polynomial having degree is 1 is called linear polynomial and at most one zeroes. 2. Quadratic polynomial —a polynomial having degree is 2 is called quadratic polynomial and at most two zeroes. 3. Cubic polynomial — a polynomial having degree is 3 is called cubic polynomial and at most three zeroes. Zeroes of