Question

Using factor theorem, factorize each of the following polynomial:
x⁴+10x³+35x²+50x +24

Answer 1

The factorised given polynomial is $x^4+10x^3+35x^2+50x+24=(x+1)(x+2)(x+3)(x+4)$

Step-by-step explanation:

• Given that the polynomial $x^4+10x^3+35x^2+50x+24$

• Since the degree of the given polynomial is 4
• Therefore it has 4 factors

To factorise the given polynomial  :

Let f(x) be the given polynomial

$f(x)=x^4+10x^3+35x^2+50x+24$

By using the Factor theorem here

• Put x=-1  in the polynomial f(x) we get

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

$=1-10+35-50+24$

$=60-60$

$=0$

$f(-1)=0$

Therefore x+1 is a factor and it satisfies the given polynomial.( by factor theorem )                                  

• Put x=-2  in the polynomial f(x) we get

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

$=16-80+140-100+24$

$=140-140$

$=0$

$f(-2)=0$

Therefore x+2 is a factor and it satisfies the given polynomial.( by factor theorem )        

• Put x=-3  in the polynomial f(x) we get

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

$=81-270+315-150+24$

$=420-420$

$=0$

$f(-3)=0$

Therefore x+3 is a factor and it satisfies the given polynomial.( by factor theorem )        

• Put x=-4  in the polynomial f(x) we get

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

$=256-640+560-200+24$

$=840-840$

$=0$

$f(-3)=0$

Therefore x+4 is a factor and it satisfies the given polynomial.( by factor theorem )        

Therefore the factors are (x+1) ,(x+2), (x+3) and (x+4)

The factorised given polynomial is $x^4+10x^3+35x^2+50x+24=(x+1)(x+2)(x+3)(x+4)$