Factor Theorem Learning

In the factor theorem if there is no remainder, it is to be naturally concluded that the given expression is completely divisible by the given divisor which means, in other words, that the divisor is a factor of the given expression. If any integral expression in x vanishes when 'a' is substituted for x, then ' x - a' is a factor of the expression.Similarly, if any integral expression in x vanishes when '-a' is substituted for x, then x+a is a factor of the expression.

Alternative form of the factor theorem:

If f(a) = 0, then x-a is a factor of f(x).

cor 1: If f(-a)= 0, then x+a is factor of f(x).

cor 2: If (-a/b)= 0, then bx+a is a factor of f(x).

cor 3: If polynomial f(x) vanishes when x= a and also x=b, then f(x) is exactly divisible by (x-a)(x-b).

Only those values of x will make f(x) zero which is factors of the constant term e.g. if f(x)= x³-19x-30, x can only have either of the values and no other values for making f(x). the reason is that constant term is always the product of the roots of the equation f(x)=0.

Examples for factor theorem learning:

1.Factorise: x³ – 7x + 6.

Solution:

Here f(x) = x³ – 7x + 6

by putting x = 1, f(1) = 1-7 + 6

=0

therefore x-1 is a factor of x³ – 7x + 6

now, x³-7x+6 = x² (x-1) + x(x-1) -6 (x-1)

= (x-1)(x² +x -6)

=(x-1)(x+3)(x-2)

2.If the expression x³ + 3x² + 4x + p contains x + 6 as a factor, find p.

Solution:

f(x)= x³ + 3x² + 4x + p

as x + 6 is its, so

f(-6) = 0.

i.e, (-6)³ + 3(-6)² + 4(-6) + p=0

i.e, -216 +108 - 24 + p=0

i.e, -132 + p=0

therefore, p=132.