Express 81-18/p+1/p² as a perfect square
Answers
Simplify ——
p2
Equation at the end of step
1
:
181
81 - ———)
p2
STEP
2
:
Rewriting the whole as an Equivalent Fraction
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using p2 as the denominator :
p p • p2
p = — = ——————
1 p2
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:
p • p2 + 1 p3 + 1
—————————— = ——————
p2 p2
Equation at the end of step
2
:
(p3 + 1)
81 - ————————
p2
STEP
3
:
p3+1
Divide 18 by ————
p2
Trying to factor as a Sum of Cubes:
3.1 Factoring: p3 + 1
Theory : A sum of two perfect cubes, a3 + b3 can be factored into :
(a+b) • (a2-ab+b2)
Proof : (a+b) • (a2-ab+b2) =
a3-a2b+ab2+ba2-b2a+b3 =
a3+(a2b-ba2)+(ab2-b2a)+b3=
a3+0+0+b3=
a3+b3
Check : 1 is the cube of 1
Check : p3 is the cube of p1
Factorization is :
(p + 1) • (p2 - p + 1)
Trying to factor by splitting the middle term
3.2 Factoring p2 - p + 1
The first term is, p2 its coefficient is 1 .
The middle term is, -p its coefficient is -1 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -1 .
-1 + -1 = -2
1 + 1 = 2
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step3
18p2
81 - ——————————————————————
(p + 1) • (p2 - p + 1)
STEP4
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using (p+1) • (p2-p+1) as the denominator :
81 81 • (p + 1) • (p2 - p + 1)
1 (p + 1) • (p2 - p + 1)
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
81 • (p+1) • (p2-p+1) - (18p2) = 81p³-18p²+81
1 • (p+1) • (p2-p+1) 1•(p+1)•( p²-p+1)
STEP5
Pulling out like terms :
5.1 Pull out like factors :
81p3 - 18p2 + 81 = 9 • (9p3 - 2p2 + 9)
Polynomial Roots Calculator :
5.2 Find roots (zeroes) of : F(p) = 9p3 - 2p2 + 9
Polynomial Roots Calculator is a set of methods aimed at finding values of p for which F(p)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers p which can be expressed as the quotient of two integers
The 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 Coefficient
In this case, the Leading Coefficient is 9 and the Trailing Constant is 9.
The factor(s) are:
of the Leading Coefficient : 1,3 ,9
of the Trailing Constant : 1 ,3 ,9
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -2.00
-1 3 -0.33 8.44
-1 9 -0.11 8.96
-3 1 -3.00 -252.00
-9 1 -9.00 -6714.00
1 1 1.00 16.00
1 3 0.33 9.11
1 9 0.11 8.99
3 1 3.00 234.00
9 1 9.00 6408.00
Polynomial Roots Calculator found no rational roots
Final result :
9 • (9p3 + 2p2 + 9)
——————————————————————
(p + 1) • (p2 + p + 1)