simplify : (2y-3)(3y-5)(y+1)
Answers
Step-by-step explanation:
Step by Step Solution:
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STEP
1
:
5
Simplify ——
y2
Equation at the end of step
1
:
5
(((2 • (y2)) - 3y) - ——) - 1
y2
STEP
2
:
Equation at the end of step
2
:
5
((2y2 - 3y) - ——) - 1
y2
STEP
3
:
Rewriting the whole as an Equivalent Fraction
3.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using y2 as the denominator :
2y2 - 3y (2y2 - 3y) • y2
2y2 - 3y = ———————— = ———————————————
1 y2
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
4
:
Pulling out like terms
4.1 Pull out like factors :
2y2 - 3y = y • (2y - 3)
Adding fractions that have a common denominator :
4.2 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:
y • (2y-3) • y2 - (5) 2y4 - 3y3 - 5
————————————————————— = —————————————
y2 y2
Equation at the end of step
4
:
(2y4 - 3y3 - 5)
——————————————— - 1
y2
STEP
5
:
Rewriting the whole as an Equivalent Fraction
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using y2 as the denominator :
1 1 • y2
1 = — = ——————
1 y2
Polynomial Roots Calculator :
5.2 Find roots (zeroes) of : F(y) = 2y4 - 3y3 - 5
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 2 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,5
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 y + 1
-1 2 -0.50 -4.50
-5 1 -5.00 1620.00
-5 2 -2.50 120.00
1 1 1.00 -6.00
1 2 0.50 -5.25
5 1 5.00 870.00
5 2 2.50 26.25
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
2y4 - 3y3 - 5
can be divided with y + 1
Polynomial Long Division :
5.3 Polynomial Long Division
Dividing : 2y4 - 3y3 - 5
("Dividend")
By : y + 1 ("Divisor")
dividend 2y4 - 3y3 - 5
- divisor * 2y3 2y4 + 2y3
remainder - 5y3 - 5
- divisor * -5y2 - 5y3 - 5y2
remainder 5y2 - 5
- divisor * 5y1 5y2 + 5y
remainder - 5y - 5
- divisor * -5y0 - 5y - 5
remainder 0
Quotient : 2y3-5y2+5y-5 Remainder: 0
Polynomial Roots Calculator :
5.4 Find roots (zeroes) of : F(y) = 2y3-5y2+5y-5
See theory in step 5.2
In this case, the Leading Coefficient is 2 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,5
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -17.00
-1 2 -0.50 -9.00
-5 1 -5.00 -405.00
-5 2 -2.50 -80.00
1 1 1.00 -3.00
1 2 0.50 -3.50
5 1 5.00 145.00
5 2 2.50 7.50
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
5.5 Adding up the two equivalent fractions
(2y3-5y2+5y-5) • (y+1) - (y2) 2y4 - 3y3 - y2 - 5
————————————————————————————— = ——————————————————
y2 y2
Checking for a perfect cube :
5.6 2y4 - 3y3 - y2 - 5 is not a perfect cube
Trying to factor by pulling out :
5.7 Factoring: 2y4 - 3y3 - y2 - 5
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -y2 - 5
Group 2: 2y4 - 3y3
Pull out from each group separately :
Group 1: (y2 + 5) • (-1)
Group 2: (2y - 3) • (y3)
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 :
5.8 Find roots (zeroes) of : F(y) = 2y4 - 3y3 - y2 - 5
See theory in step 5.2
In this case, the Leading Coefficient is 2 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,5
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -1.00
-1 2 -0.50 -4.75
-5 1 -5.00 1595.00
-5 2 -2.50 113.75
1 1 1.00 -7.00
1 2 0.50 -5.50
5 1 5.00 845.00
5 2 2.50 20.00
Polynomial Roots Calculator found no rational roots
Final result :
2y4 - 3y3 - y2 - 5
——————————————————
y2
Answer:
(2y-3)(3y-5)(y+1)=(6y^2-9y-10y+15)(y+1)
=(6y^2-19y+15)(y+1)
=6y^3-19y^2+15y+6y^2-9y+15
(2y-3)(3y-5)(y+1)=6y^3-13y^2+6y+15