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What is factor theorem |
| Answer» If f(x) is a polynomial of degree\xa0n ≥ 1\xa0and ‘a’ is any real number, then\t(x-a) is a factor of f(x) , if f(a)=0\tIts converse “ if (x-a) is a factor of the polynomial f(x), then f(a)=0”In mathematics, factor theorem is used as a linking factor and zeros of the\xa0polynomial. Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial equation.Steps to Use Factor TheoremStep 1 : If f(-c)=0, ( x+ c) is a factor of the polynomial f(x).Step 2 : If p(d/c)= 0, (cx-d) is a factor of the polynomial f(x).Step 3 : If p(-d/c)= 0, (cx+d) is a factor of the polynomial f(x).Step 4 : If p(c)=0 and p(d) =0, then (x-c) and (x-d) is a factor of the polynomial.Rather than finding the factors by using polynomial long division method, the best way to find the factors are factor theorem and synthetic division method. The factor theorem is mainly used to remove the known zeros from polynomials leaving all unknown zeros unimpaired, thus by finding the zeros easily to produce the lower degree polynomial.Example:Consider the polynomial function f(x)= x2\xa0+2x -15The values of x for which f(x)=0 are called the roots of the function. By solving the equation, f(x)=0Then, we getx2\xa0+2x -15 =0(x+5)(x-3)=0(x+5)=0 or (x-3)=0x = -5 or x = 3Because (x+5) and (x-3) is a factor of x2\xa0+2x -15, -5 and 3 are the solutions to the equation x2\xa0+2x -15=0, we can also check as follows:If x = -5 is the solution , thenf(x)= x2\xa0+2x -15f(-5) = (-5)2\xa0+ 2(-5) – 15f(-5) = 25-10-15f(-5)=25-25f(-5)=0If x=3 is the solution, themf(x)= x2\xa0+2x -15f(3)= 32\xa0+2(3) – 15f(3) = 9 +6 -15f(3) = 15-15f(3)= 0If the remainder is zero, (x-c) is a polynomial of f(x) | |