Add 4y(3y2-5y-7) and 2(y3-4y2+5)
Answers
Answer:
-4y(3y2+5y-7)+2(y3-4y2+5)
Final result :
-2 • (5y2 + 19y + 5) • (y - 1)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(0-(4y•(((3•(y2))+5y)-7)))+(2•(((y3)-22y2)+5))
Step 2 :
Polynomial Roots Calculator :
2.1 Find roots (zeroes) of : F(y) = y3-4y2+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 1 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1
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
-5 1 -5.00 -220.00
1 1 1.00 2.00
5 1 5.00 30.00
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
y3-4y2+5
can be divided with y+1
Polynomial Long Division :
2.2 Polynomial Long Division
Dividing : y3-4y2+5
("Dividend")
By : y+1 ("Divisor")
dividend y3 - 4y2 + 5
- divisor * y2 y3 + y2
remainder - 5y2 + 5
- divisor * -5y1 - 5y2 - 5y
remainder 5y + 5
- divisor * 5y0 5y + 5
remainder 0
Quotient : y2-5y+5 Remainder: 0
Trying to factor by splitting the middle term
2.3 Factoring y2-5y+5
The first term is, y2 its coefficient is 1 .
The middle term is, -5y its coefficient is -5 .
The last term, "the constant", is +5
Step-1 : Multiply the coefficient of the first term by the constant 1 • 5 = 5
Step-2 : Find two factors of 5 whose sum equals the coefficient of the middle term, which is -5 .
-5 + -1 = -6
-1 + -5 = -6
1 + 5 = 6
5 + 1 = 6
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 2 :
(0-(4y•(((3•(y2))+5y)-7)))+2•(y2-5y+5)•(y+1)
Step 3 :
Equation at the end of step 3 :
(0-(4y•((3y2+5y)-7)))+2•(y2-5y+5)•(y+1)
Step 4 :
Trying to factor by splitting the middle term
4.1 Factoring 3y2+5y-7
The first term is, 3y2 its coefficient is 3 .
The middle term is, +5y its coefficient is 5 .
The last term, "the constant", is -7
Step-1 : Multiply the coefficient of the first term by the constant 3 • -7 = -21
Step-2 : Find two factors of -21 whose sum equals the coefficient of the middle term, which is 5 .
-21 + 1 = -20
-7 + 3 = -4
-3 + 7 = 4
-1 + 21 = 20
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 4 :
(0-4y•(3y2+5y-7))+2•(y2-5y+5)•(y+1)
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
-10y3 - 28y2 + 28y + 10 =
-2 • (5y3 + 14y2 - 14y - 5)
Checking for a perfect cube :
6.2 5y3 + 14y2 - 14y - 5 is not a perfect cube
Trying to factor by pulling out :
6.3 Factoring: 5y3 + 14y2 - 14y - 5
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 5y3 - 5
Group 2: 14y2 - 14y
Pull out from each group separately :
Group 1: (y3 - 1) • (5)
Group 2: (y - 1) • (14y)
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) = 5y3 + 14y2 - 14y - 5
See theory in step 2.1
In this case, the Leading Coefficient is 5 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,5
of the Trailing Constant : 1 ,5
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 18.00
-1 5 -0.20 -1.68
-5 1 -5.00 -210.00
1 1 1.00 0.00 y - 1
1 5 0.20 -7.20
5 1 5.00 900.00
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
5y3 + 14y2 - 14y - 5
can be divided with y - 1
Polynomial Long Division :
6.5 Polynomial Long Division
Dividing : 5y3 + 14y2 - 14y - 5
("Dividend")
By : y - 1 ("Divisor")
dividend 5y3 + 14y2 - 14y - 5
- divisor * 5y2 5y3 - 5y2
remainder 19y2 - 14y - 5
- divisor * 19y1 19y2 - 19y
remainder 5y - 5
- divisor * 5y0 5y - 5
remainder 0
Quotient : 5y2+19y+5 Remainder: 0
Trying to factor by splitting the middle term
6.6 Factoring 5y2+19y+5
The first term is, 5y2 its coefficient is 5 .
The middle term is, +19y its coefficient is 19 .
The last term, "the constant", is +5
Step-1 : Multiply the coefficient of the first term by the constant 5 • 5 = 25
Step-2 : Find two factors of 25 whose sum equals the coefficient of the middle term, which is 19 .
-25 + -1 = -26
-5 + -5 = -10
-1 + -25 = -26
1 + 25 = 26
5 + 5 = 10
25 + 1 = 26
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
-2 • (5y2 + 19y + 5) • (y - 1)
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