Question

factorise the cubic polynomial p(x)=x^3+3x^2+5x+15

Answer 1

Question

Factorise the cubic polynomial

p(x)= x³ +3x² +5x +15

Solution:

By using regrouping method

We have,

---> x³ +3x² +5x +15

Taking common

==> x²(x+3)+5(x+3)

==>(x²+5)(x+3)

Hence, the factors are (x²+5) & (x+3)

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Factoring using hit and trial method

Let p(x)= x³+3x²+5x+15

and g(x) = x +1 is the factor of p(x) then by factor theorem,

x+1 =0

x = -1

So,

p(-1) = (-1)³ +3(-1)² +5(-1) +15

= -1 +3 -5 +15

= 12 not equal to 0

so it's not the factor of p(x)

Let g(x) = x +2

p(-2) = (-2)³ +3(-2)² +5(-2)+15

= -8 +12-10 +15

= 9 which is nt equal to 0

Now

lets take (x+3)

so,

p(-3)= (-3)³ +3(-3)² +5(-3) +15

= -27 + 27 -15 +15

= 0

Hence (x+3) is a factor of p(x).

Now,

we will divide p(x) by g(x)

x²+5

x+3/x³+3x²+5x+15\_

x³+3x²

(-) (-) (both cancelled)

5x +15

(-) (-) (cancelled)

0

Since, x+3 is a factor of p(x)

on dividing we get

x² + 5

Hence, x² +5 is also a factor of p(x) as remaimder is 0.

Thus, our factors are (x+3)(x²+5)