Question

factories using factor theorem :
1. x³-6x²+11x-6
2. x³-3x²-4x+12
3. x³-19x-30
please help me​

Answer 1

1. x3−6x2+11x−6

=x3−5x2−x2+6x+5x−6

=x3−5x2+6x−x2+5x−6

=x(x2−5x+6)−1(x2−5x+6)

=(x−1)(x2−5x+6)

=(x−1)(x2−2x−3x+6)

=(x−1)[x(x−2)−3(x−2)]

=(x−1)(x−2)(x−3)

2.x3−3x2+4x−12

=x2(x−3)+4(x−3)

=(x2+4)(x−3)

∴(x−3) is a factor of x3−3x2+4x−12.

3.Let us assume x=−2,

Given, f(x)=x3–19x–30

Now, substitute the value of x in f(x),

f(−1)=(−2)3–19(−2)–30

=−8+38–30

=−38+38

=0

Therefore, x+2 is the factor of x3–19x–30.

Now, dividing x3–19x–30 by x+2 we get,

Therefore, x3–19x–30=(x+2)(x2–2x–15)

=(x+2)(x2–5x+3x–15)

=(x+2)(x–5)(x+3)