factorise mn(p^2+q^2)-m^2pq-n^2pq
Answers
((p2)-(q2)) (((m2)+2mn)+(n2))
———————————•—————————————————
((m2)-(n2)) ((3p2+2pq)-q2)
STEP
2
:
m2 + 2mn + n2
Simplify ——————————————
3p2 + 2pq - q2
Trying to factor a multi variable polynomial :
2.1 Factoring m2 + 2mn + n2
Try to factor this multi-variable trinomial using trial and error
Found a factorization : (m + n)•(m + n)
Detecting a perfect square :
2.2 m2 +2mn +n2 is a perfect square
It factors into (m+n)•(m+n)
which is another way of writing (m+n)2
How to recognize a perfect square trinomial:
• It has three terms
• Two of its terms are perfect squares themselves
• The remaining term is twice the product of the square roots of the other two terms
Trying to factor a multi variable polynomial :
2.3 Factoring 3p2 + 2pq - q2
Try to factor this multi-variable trinomial using trial and error
Found a factorization : (3p - q)•(p + q)
Equation at the end of step
2
:
((p2)-(q2)) (m+n)2
———————————•————————————
((m2)-(n2)) (3p-q)•(p+q)
STEP
3
:
p2 - q2
Simplify ———————
m2 - n2
Trying to factor as a Difference of Squares:
3.1 Factoring: p2 - q2
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : p2 is the square of p1
Check : q2 is the square of q1
Factorization is : (p + q) • (p - q)
Trying to factor as a Difference of Squares:
3.2 Factoring: m2 - n2
Check : m2 is the square of m1
Check : n2 is the square of n1
Factorization is : (m + n) • (m - n)
Equation at the end of step
3
:
(p + q) • (p - q) (m + n)2
————————————————— • ——————————————————
(m + n) • (m - n) (3p - q) • (p + q)
STEP
4
:
Canceling Out
4.1 Cancel out (p+q) which appears on both sides of the fraction line.
Dividing Exponential Expressions:
4.2 Divide (m+n)2 by (m+n)
The rule says : To divide exponential expressions which have the same base, subtract their exponents.
In our case, the common base is (m+n) and the exponents are :
2
and 1 , as (m+n) is the same number as (m+n)1
The quotient is therefore, (m+n)(2-1) = (m+n)1
Omit the '1' in the exponent altogether. Anything to the first power is the number itself so there is usually no reason to write down the '1
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