Using factor theorem, factorize of the polynomials:

Solution:

Given, $f(x)=x^{4}+10 x^{3}+35 x^{2}+50 x+24$

The constant term in f(x) is equal to 24

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

Let, x + 1 = 0

=> x = -1

Substitute the value of x in f(x)

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

= 1-10 + 35 - 50 + 24

= 0

=> (x + 1) is the factor of f(x)

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

Since, f(x) is a polynomial of degree 4, it cannot have more than four linear factors.

=> f(x) = k(x + 1)(x + 2)(x + 3)(x + 4)

$=x^{4}+10 x^{3}+35 x^{2}+50 x+24=k(x+1)(x+2)(x+3)(x+4)$

Substitute x = 0 on both sides

=> 0 + 0 + 0 + 0 + 24 = k(1)(2)(3)(4)

=> 24 = k(24)

=> k = 1

Substitute k = 1 in f(x) = k(x + 1)(x + 2)(x + 3)(x + 4)

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

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

hence, $x^{4}+10 x^{3}+35 x^{2}+50 x+24=(x+1)(x+2)(x+3)(x+4)$