Adding, Subtracting And Finding The Least Common Multiple

Step 1 :

n Simplify ———————————— 11 - n - 3n2

Step 2 :

Pulling out like terms :

2.1 Pull out like factors : 11 - n - 3n2 = -1 • (3n2 + n - 11)

Trying to factor by splitting the middle term

2.2 Factoring 3n2 + n - 11 The first term is, 3n2 its coefficient is 3 .The middle term is, +n its coefficient is 1 .The last term, "the constant", is -11 Step-1 : Multiply the coefficient of the first term by the constant 3 • -11 = -33 Step-2 : Find two factors of -33 whose sum equals the coefficient of the middle term, which is 1 .

-33	+	1	=	-32
-11	+	3	=	-8
-3	+	11	=	8
-1	+	33	=	32

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

Equation at the end of step 2 :

(3n + 5) n ———————— - (5n • —————————————) (2n - 7) -3n2 - n + 11

Step 3 :

Step 4 :

Pulling out like terms :

4.1 Pull out like factors : -3n2 - n + 11 = -1 • (3n2 + n - 11)

Multiplying exponential expressions :

4.2 n1 multiplied by n1 = n(1 + 1) = n2

Equation at the end of step 4 :

(3n + 5) 5n2 ———————— - ————————————— (2n - 7) -3n2 - n + 11

Step 5 :

3n + 5 Simplify —————— 2n - 7

Equation at the end of step 5 :

(3n + 5) 5n2 ———————— - ————————————— 2n - 7 -3n2 - n + 11

Step 6 :

Step 7 :

Pulling out like terms :

7.1 Pull out like factors : -3n2 - n + 11 = -1 • (3n2 + n - 11)

Calculating the Least Common Multiple :

7.2 Find the Least Common Multiple The left denominator is : 2n - 7 The right denominator is : -3n2 - n + 11

Number of times each Algebraic Factor appears in the factorization of:

Algebraic Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
2n - 7	1	0	1
-3n2 - n + 11	0	1	1

Least Common Multiple: (2n - 7) • (-3n2 - n + 11)

Calculating Multipliers :

7.3 Calculate multipliers for the two fractions Denote the Least Common Multiple by L.C.M Denote the Left Multiplier by Left_M Denote the Right Multiplier by Right_M Denote the Left Deniminator by L_Deno Denote the Right Multiplier by R_Deno Left_M = L.C.M / L_Deno = -3n2 - n + 11 Right_M = L.C.M / R_Deno = 2n - 7

Making Equivalent Fractions :

7.4 Rewrite the two fractions into equivalent fractionsTwo fractions are called equivalent if they have the same numeric value. For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well. To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L. Num. (3n+5) • (-3n2-n+11) —————————————————— = ———————————————————— L.C.M (2n-7) • (-3n2-n+11) R. Mult. • R. Num. 5n2 • (2n-7) —————————————————— = ———————————————————— L.C.M (2n-7) • (-3n2-n+11)

Adding fractions that have a common denominator :

7.5 Adding up the two equivalent fractions Add the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

(3n+5) • (-3n2-n+11) - (5n2 • (2n-7)) -19n3 + 17n2 + 28n + 55 ————————————————————————————————————— = —————————————————————————— (2n-7) • (-3n2-n+11) (2n - 7) • (-3n2 - n + 11)

Checking for a perfect cube :

7.6 -19n3 + 17n2 + 28n + 55 is not a perfect cube

Trying to factor by pulling out :

7.7 Factoring: -19n3 + 17n2 + 28n + 55 Thoughtfully split the expression at hand into groups, each group having two terms :Group 1: 28n + 55 Group 2: -19n3 + 17n2 Pull out from each group separately :Group 1: (28n + 55) • (1)Group 2: (19n - 17) • (-n2)Bad news !! Factoring by pulling out fails : The groups have no common factor and can not be added up to form a multiplication.

Step 8 :

Pulling out like terms :

8.1 Pull out like factors : -3n2 - n + 11 = -1 • (3n2 + n - 11)

Polynomial Roots Calculator :

8.2 Find roots (zeroes) of : F(n) = -19n3 + 17n2 + 28n + 55Polynomial Roots Calculator is a set of methods aimed at finding values of n for which F(n)=0 Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers n which can be expressed as the quotient of two integersThe Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is -19 and the Trailing Constant is 55. The factor(s) are: of the Leading Coefficient : 1,19 of the Trailing Constant : 1 ,5 ,11 ,55 Let us test ....

P	Q	P/Q	F(P/Q)	Divisor
-1	1	-1.00	63.00
-1	19	-0.05	53.58
-5	1	-5.00	2715.00
-5	19	-0.26	49.16
-11	1	-11.00	27093.00
-11	19	-0.58	48.17
-55	1	-55.00	3211065.00
-55	19	-2.89	577.27
1	1	1.00	81.00
1	19	0.05	56.52
5	1	5.00	-1755.00
5	19	0.26	63.20
11	1	11.00	-22869.00
11	19	0.58	73.22
55	1	55.00	-3108105.00
55	19	2.89	-182.37

Polynomial Roots Calculator found no rational roots

Final result :

+19n3 + 17n2 + 28n + 55 ————————————————————————— (7 + 2n) • (3n2 + n + 11)