Factorise
mn+m+n+l
please explain in detail
Answers
(m+n/m-n)+(m-n/m+n) Final result : 2m2 ——— m
Step by step solution :Step 1 : n Simplify — m Equation at the end of step 1 : n n ((m+—)-n)+((m-—)+n) m m Step 2 :Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using m as the denominator :
m m • m m = — = ————— 1 m
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
2.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
m • m - (n) m2 - n ——————————— = —————— m m Equation at the end of step 2 : n (m2 - n) ((m + —) - n) + (———————— + n) m m Step 3 :Rewriting the whole as an Equivalent Fraction :
3.1 Adding a whole to a fraction
Rewrite the whole as a fraction using m as the denominator :
n n • m n = — = ————— 1 m Trying to factor as a Difference of Squares :
3.2 Factoring: m2 - n
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 : m2 is the square of m1
Check : n1 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares
Adding fractions that have a common denominator :
3.3 Adding up the two equivalent fractions
(m2-n) + n • m m2 + mn - n —————————————— = ——————————— m m Equation at the end of step 3 : n (m2 + mn - n) ((m + —) - n) + ————————————— m m Step 4 : n Simplify — m Equation at the end of step 4 : n (m2 + mn - n) ((m + —) - n) + ————————————— m m Step 5 :Rewriting the whole as an Equivalent Fraction :
5.1 Adding a fraction to a whole
Rewrite the whole as a fraction using m as the denominator :
m m • m m = — = ————— 1 m Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
m • m + n m2 + n ————————— = —————— m m Equation at the end of step 5 : (m2 + n) (m2 + mn - n) (———————— - n) + ————————————— m m Step 6 :Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using m as the denominator :
n n • m n = — = ————— 1 m Adding fractions that have a common denominator :
6.2 Adding up the two equivalent fractions
(m2+n) - (n • m) m2 - mn + n ———————————————— = ——————————— m m Equation at the end of step 6 : (m2 - mn + n) (m2 + mn - n) ————————————— + ————————————— m m Step 7 :Trying to factor a multi variable polynomial :
7.1 Factoring m2 - mn + n
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Trying to factor a multi variable polynomial :
7.2 Factoring m2 + mn - n
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Adding fractions which have a common denominator :
7.3 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
(m2-mn+n) + (m2+mn-n) 2m2 ————————————————————— = ——— m m Final result : 2m2 ——— m
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