You can put this solution on YOUR website! 1. (x-6)(9x+4)(9x-4) 2. (x-1)(x+2)-(x+3)(x-4)
(x-6)(9x+4)(9x-4)
=(x-6)[(9x+4)(9x-4)]
=(x-6)[(9x)^2-(4)^2]
=(x-6)[81x^2-16]
(using (a+b)(a-b) = (a^2-b^2) formula and here a=9x and b = 4 )
=x(81x^2-16)-6(81x^2-16)
[product expansion using the rule: first term of the first bracket (along with the sign)multiplied by the whole of the next bracket PLUS second term of the first bracket (along with the sign) multiplied by the whole of the next bracket and so on until all the terms in the first bracket are exhausted)
=81x^3-16x-486x^2+96
=81x^3-486x^2-16x+96 (using additive commutativity- what terms?!: -16x-486x^2 = -486x^2-16x )
Note: Why is the above commutativity performed instead of leaving the expansion as it is?
Here it is done to present the answer with the terms in the descending order of powers of x.
Note: Normally the multiplication is done as follows:
(x-6)(9x+4)(9x-4)
=[(x-6)(9x+4)](9x-4)
=[x(9x+4)-6(9x+4)](9x-4)
=(9x^2+4x-54x-24)(9x-4)
=(9x^2-50x-24)(9x-4)
=9x^2(9x-4)-50x(9x-4)-24(9x-4)
=81x^3-36x^2-450x^2+200x-216x+96
=81x^3-486x^2-16x+96
A little more number of steps and hence longer time taken for completing the problem
Whenever a bonus feature like (a+b)(a-b) that gives quickly (a^2-b^2) appears in a problem, we must make the best use of it to cut short time as well as to make the problem steps look more elegant.
Note: In the first illustration the product ABC is seen as A(BC)
and in the second illustration the product ABC is seen as (AB)C
And A(BC) = (AB)C which is infact associativity w.r.t.multiplication.
2) (x-1)(x+2)-(x+3)(x-4)
=[x(x+2)-1(x+2)]-[(x+3)(x-4)]
=(x^2+2x-x-2)-[x(x-4)+3(x-4)]
=x^2+x-2-[x^2-4x+3x-12]
=x^2+x-2-[x^2-x-12]
=x^2+x-2-x^2+x+12
=(x^2-x^2)+(x+x)+(-2+12)
=0+2x+10
=2x+10
=2(x+5)
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