Factorize
(x+1)(x+2)(x+3)(x+6)-3x^2
carefully expand to get
x^4+12x^3+47x^2+72x+36 - 3x^2
= x^4 + 12x^3 + 44x^2 + 72x + 36
I assume it will factor, (or else why have the question?)
I tried all possible factors of 36, and none made the expression zero, so there are no binary factors with rational numbers.
I then assumed we had something like
(x^2 + bx + c)(x^2 + dx + e), expanded that , then compared coefficients.
solving these 4 equations in 4 unknowns gave me
b= 4
c = 6
d = 8
e = 6
for:
(x^2 + 4x + 6)(x^2 + 8x + 6)
Nasty question.
Nice solution i was very bore due to this ques
I am not happy to learn that I went through the effort of solving the question to relieve your boredom.
To factorize the expression (x+1)(x+2)(x+3)(x+6)-3x^2, we can start by simplifying the expression inside the parentheses.
First, let's expand the first part of the expression:
(x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2
Next, we can expand the second part of the expression:
(x^2 + 3x + 2)(x+3) = x^3 + 3x^2 + 2x + 3x^2 + 9x + 6 = x^3 + 6x^2 + 11x + 6
Lastly, we multiply the result by (x+6):
(x^3 + 6x^2 + 11x + 6)(x+6) = x^4 + 6x^3 + 11x^2 + 6x + 6x^3 + 36x^2 + 66x + 36
Combine like terms:
x^4 + 12x^3 + 47x^2 + 72x + 36
Now, let's subtract 3x^2 from the expanded expression:
x^4 + 12x^3 + 47x^2 + 72x + 36 - 3x^2
Combine like terms again:
x^4 + 12x^3 + 44x^2 + 72x + 36
Therefore, the given expression (x+1)(x+2)(x+3)(x+6)-3x^2 can be written as x^4 + 12x^3 + 44x^2 + 72x + 36 after expanding and simplifying.