Question

Q) Factorize x⁴ + x
Q) Factorize (m+2n)² + 101(m+2n) + 100

Answer 1

x⁴-x=

Pulling out like terms :

2.1 Pull out like factors :

x⁴ - x = x • (x³ - 1)

Trying to factor as a Difference of Cubes:

2.2 Factoring: x³ - 1

Theory : A difference of two perfect cubes, a³ - b³ can be factored into
(a-b) • (a² +ab +b²)

Proof : (a-b)•(a²+ab+b²) =
a³+a²b+ab²-ba²-b²a-b³ =
a³+(a²b-ba²)+(ab²-b²a)-b³ =
a³+0+0+b³ =
a³+b³

Check : 1 is the cube of 1
Check : x³ is the cube of x¹

Factorization is :
(x - 1) • (x² + x + 1)

Trying to factor by splitting the middle term

2.3 Factoring x² + x + 1

The first term is, x² its coefficient is 1 .
The middle term is, +x its coefficient is 1 .
The last term, "the constant", is +1

Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1

Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is 1 .

	-1	+	-1	=	-2
	1	+	1	=	2

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result : x • (x - 1) • (x² + x + 1) (m+2n)² + 101(m+2n) + 100 Put m+2n=x then we get equation is x^2+101x+100=0 x^2+100x+x+100=0 x(x+100)+1(x+100)=0 (x+100)(x+1)=0 now put the value of x in above equation (m+2n+100)(m+2n+1)=0