Question

11. Using the Factor Theorem, factorise the following:
(a) x4 + 10x3 + 35x2 + 50x + 24
(b) 244-713 - 13y2 + 63y - 45​

Answer 1

Answer:

Step-by-step explanation:

Let f(x) = x4 + 10x3 + 35x2 + 50x + 24  

Constant term = 24

 Factors of 24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24

Let x + 1 = 0 or x = -1  

f(-1) = (-1)4 + 10(-1)3 + 35(-1)2 + 50(-1) + 24

= 1 – 10 + 35 – 50 + 24

 = 0

f(1) = 0  

(x + 1) is a factor of f(x)  

Likewise, (x + 2),(x + 3),(x + 4) are also the factors of f(x)

Hence f(x) = (x + 1) (x + 2)(x + 3)(x + 4)

Answer 2

Answer:

(a) (x + 1)

(x + 2)

(x + 4)

(x + 4)

Step-by-step explanation:

x4 + 10x3 + 35x2 + 50x + 24

As the last term is 24  x-2 or x+2 could be factors

By the Factor theorem

f(-2) = 0 if x+2 is a factor.

f(-2) = (-2)^4 + 10)(-2)^3 + 35(-2)^2 + 50*-2 + 24

= 16  - 80 + 140 - 100 + 24

= 180 - 180

= 0.

So (x + 2) is a factor.

Now try f(-1) , f(1).

f(-1) = 1 - 10 + 35 - 50 + 24

= 60 - 60 = 0

so (x + 1) is a factor)

No do long multiplication:

(x + 2)(x + 1) = x^2 + 3x + 2.

x^2 + 3x + 2) x4 + 10x3 + 35x2 + 50x + 24(x^2 + 7x + 12<--- Quotient

                -    x4 + 3x^3 + 2x^2

                             7x^3  + 33x^2 + 50x

                         -   7x^3  + 21x^2 + 14x

                                          12x^2 + 36x + 24

                                           12x^2 + 36x + 24

                         

x^2 + 7x + 12 = (x + 3)(x + 4).