Factoring ax^2+bx+c Factoring ax^2+bx+c First we will focus on quadratic polynomial. A 2nd degree polynomial which is of the form p(x) = ax^2 + bx + c, where we have a ≠0, is called a quadratic polynomial. Before learning how to solve any quadratic polynomial, we will first understand the following related terms. Value of a polynomial at a given point: If we have any polynomial say p(x) in x and if n is any real number, then the value obtained by putting x = n in the polynomial p(x) at x=n is the value of the polynomial at a given point n. To find ZERO of the Polynomial: Any given number n is called the ZERO of the polynomial p(x), if we get p(n) = 0. To understand it more clearly let us take a polynomial p(x)= x^2 -2x -3. Find the value of the polynomial at x= 3 and x= -1 On putting x= 3, we get: p( 3) = 3^2 - 2* 3 -3 =9-6-3 Know More About :- The Substitution Method
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On putting x = -1, we get: p(-1) = (-1) ^2 - 2 *(-1) -3 = (1) +2 - 3 =3-3 =0 In both the above cases we come to the conclusion that p(3) and p(-1) results to zero. so 3 and -1 are zeros of the polynomial p(x). Relation between the zeros and coefficients of a quadratic polynomial.Let α and β are any two zeros of a given quadratic equation which is in the form of p(x) = ax^2 + bx + c, where we have a ≠ 0. Thus we come to the conclusion that (x-α) and (x - β) are the factors of the polynomial p(x). So, we get p(x) = (x-α) * (x - β)* k, where k is any constant value. On solving the above equation we get =9-9=0 p(x)= k. x^2 - ( α + β) x + α * β Solving further we get p(x)= k. x^2 - k.( α + β) x +k* α * β Comparing the above value of p(x) we get: a= k, b = - k.( α + β) and c =k* α *β But a = k, so b = - a.( α + β) and c =a* α *β -b/a = ( α + β) and c/a = α *β -b/a = sum of roots and c/a = product of roots Thus we conclude that we can find the sum of the roots and the product of the given quadratic equation of the given quadratic equation by the above formulae. E.g. For the given polynomial f(x) = x^2 + 7x + 12, find the sum and the product of the roots. We observe that in above equation a= 1, b= 7 and c = 12 So, sum of the roots = -b/a = -7/1 = -7 And the product of the roots = c/a = 12/1 = 12 Read More About :-Substitution Method Algebra
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