y
Step-by-step explanation:
2•3y3) - 22y2) - 3y) - —) - 2
y
STEP
4
:
Rewriting the whole as an Equivalent Fraction
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using y as the denominator :
6y3 - 4y2 - 3y (6y3 - 4y2 - 3y) • y
6y3 - 4y2 - 3y = —————————————— = ————————————————————
1 y
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
STEP
5
:
Pulling out like terms
5.1 Pull out like factors :
6y3 - 4y2 - 3y = y • (6y2 - 4y - 3)
Trying to factor by splitting the middle term
5.2 Factoring 6y2 - 4y - 3
The first term is, 6y2 its coefficient is 6 .
The middle term is, -4y its coefficient is -4 .
The last term, "the constant", is -3
Step-1 : Multiply the coefficient of the first term by the constant 6 • -3 = -18
Step-2 : Find two factors of -18 whose sum equals the coefficient of the middle term, which is -4 .
-18 + 1 = -17
-9 + 2 = -7
-6 + 3 = -3
-3 + 6 = 3
-2 + 9 = 7
-1 + 18 = 17
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
5.3 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:
y • (6y2-4y-3) • y - (6) 6y4 - 4y3 - 3y2 - 6
———————————————————————— = ———————————————————
y y
Equation at the end of step
5
:
(6y4 - 4y3 - 3y2 - 6)
————————————————————— - 2
y
STEP
6
:
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using y as the denominator :
2 2 • y
2 = — = —————
1 y
Checking for a perfect cube :
6.2 6y4 - 4y3 - 3y2 - 6 is not a perfect cube
Trying to factor by pulling out :
6.3 Factoring: 6y4 - 4y3 - 3y2 - 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -3y2 - 6
Group 2: 6y4 - 4y3
Pull out from each group separately :
Group 1: (y2 + 2) • (-3)
Group 2: (3y - 2) • (2y3)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
6.4 Find roots (zeroes) of : F(y) = 6y4 - 4y3 - 3y2 - 6
Polynomial Roots Calculator is a set of methods aimed at finding values of y for which F(y)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y 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 6 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 1.00
-1 2 -0.50 -5.88
-1 3 -0.33 -6.11
-1 6 -0.17 -6.06
-2 1 -2.00 110.00
Note - For tidiness, printing of 13 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
(6y4-4y3-3y2-6) - (2 • y) 6y4 - 4y3 - 3y2 - 2y - 6
————————————————————————— = ————————————————————————
y y
Polynomial Roots Calculator :
6.6 Find roots (zeroes) of : F(y) = 6y4 - 4y3 - 3y2 - 2y - 6
See theory in step 6.4
In this case, the Leading Coefficient is 6 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 3.00
-1 2 -0.50 -4.88
-1 3 -0.33 -5.44
-1 6 -0.17 -5.73
-2 1 -2.00 114.00
Note - For tidiness, printing of 13 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Final result :
6y4 - 4y3 - 3y2 - 2y - 6
————————————————————————
y
