p^3−10p^2+1
—————— We find roots of zeros F(p) = p^3 - 10p^2 + 1 and see there
p^2 are no rational roots
Step-by-step explanation:
p^2
Simplify ——
p^2
1.1 Canceling out p^2 as it appears on both sides of the fraction line
Equation at the end of step 1
:1
((————-(4•1))+p)-6
(p^2)
STEP 2: working left to right
1
Simplify ——
p^2
Equation at the end of step 2:
1 /p^2 ((—— - 4) + p) - 6
STEP 3:
Rewriting the whole as an Equivalent Fraction
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using p^2 as the denominator :
4 4 • p^2
4 = — = ——————
1 p^2
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 :
3.2 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:
1 - (4 • p^2) 1 - 4p^2
———————————— = ———————
p^2 p^2
Equation at the end of step 3:
(1 - 4p^2)
(————————— + p) - 6
p^2
STEP 4:
Rewriting the whole as an Equivalent Fraction
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using p2 as the denominator :
p p • p^2
p = — = ——————
1 p^2
Trying to factor as a Difference of Squares:
4.2 Factoring: 1 - 4p^2
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 : 1 is the square of 1
Check : 4 is the square of 2
Check : p^2 is the square of p^1
Factorization is : (1 + 2p) • (1 - 2p)
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
(2p+1) • (1-2p) + p • p^2 p^3 - 4p^2 + 1
———————————————————————— = ————————————
p^2 p^2
Equation at the end of step
4:
(p^3 - 4p^2 + 1)
—————————————— - 6
p^2
STEP 5:
Rewriting the whole as an Equivalent Fraction
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using p^2 as the denominator :
6 6 • p^2
6 = — = ——————
1 p^2
Polynomial Roots Calculator :
5.2 Find roots (zeroes) of : F(p) = p^3 - 4p^2 + 1
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 1 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -4.00
1 1 1.00 -2.00
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
5.3 Adding up the two equivalent fractions
(p3-4p2+1) - (6 • p2) p3 - 10p2 + 1
————————————————————— = —————————————
p2 p2
Polynomial Roots Calculator :
5.4 Find roots (zeroes) of : F(p) = p3 - 10p2 + 1
See theory in step 5.2
In this case, the Leading Coefficient is 1 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -10.00
1 1 1.00 -8.00
Polynomial Roots Calculator found no rational roots
Final result :
p3 - 10p2 + 1
—————————————
p2
