(2t - 1) • (2t - 3)
Step-by-step explanation:
STEP
5
:
12t4 - 4t3 - 39t2 + 31t - 6
Simplify ———————————————————————————
3t2 + 5t - 2
Polynomial Roots Calculator :
5.1 Find roots (zeroes) of : F(t) = 12t4 - 4t3 - 39t2 + 31t - 6
Polynomial Roots Calculator is a set of methods aimed at finding values of t for which F(t)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers t 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 12 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,4 ,6 ,12
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -60.00
-1 2 -0.50 -30.00
-1 3 -0.33 -20.37
-1 4 -0.25 -16.08
-1 6 -0.17 -12.22
-1 12 -0.08 -8.85
-2 1 -2.00 0.00 t + 2
-2 3 -0.67 -40.44
-3 1 -3.00 630.00
-3 2 -1.50 -66.00
-3 4 -0.75 -45.70
-6 1 -6.00 14820.00
1 1 1.00 -6.00
1 2 0.50 0.00 2t - 1
1 3 0.33 0.00 3t - 1
1 4 0.25 -0.70
1 6 0.17 -1.93
1 12 0.08 -3.69
2 1 2.00 60.00
2 3 0.67 -1.48
3 1 3.00 600.00
3 2 1.50 0.00 2t - 3
3 4 0.75 -2.58
6 1 6.00 13464.00
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
12t4 - 4t3 - 39t2 + 31t - 6
can be divided by 4 different polynomials,including by 2t - 3
Polynomial Long Division :
5.2 Polynomial Long Division
Dividing : 12t4 - 4t3 - 39t2 + 31t - 6
("Dividend")
By : 2t - 3 ("Divisor")
dividend 12t4 - 4t3 - 39t2 + 31t - 6
- divisor * 6t3 12t4 - 18t3
remainder 14t3 - 39t2 + 31t - 6
- divisor * 7t2 14t3 - 21t2
remainder - 18t2 + 31t - 6
- divisor * -9t1 - 18t2 + 27t
remainder 4t - 6
- divisor * 2t0 4t - 6
remainder 0
Quotient : 6t3+7t2-9t+2 Remainder: 0
Polynomial Roots Calculator :
5.3 Find roots (zeroes) of : F(t) = 6t3+7t2-9t+2
See theory in step 5.1
In this case, the Leading Coefficient is 6 and the Trailing Constant is 2.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 12.00
-1 2 -0.50 7.50
-1 3 -0.33 5.56
-1 6 -0.17 3.67
-2 1 -2.00 0.00 t+2
-2 3 -0.67 9.33
1 1 1.00 6.00
1 2 0.50 0.00 2t-1
1 3 0.33 0.00 3t-1
1 6 0.17 0.72
2 1 2.00 60.00
2 3 0.67 0.89
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
6t3+7t2-9t+2
can be divided by 3 different polynomials,including by 3t-1
Polynomial Long Division :
5.4 Polynomial Long Division
Dividing : 6t3+7t2-9t+2
("Dividend")
By : 3t-1 ("Divisor")
dividend 6t3 + 7t2 - 9t + 2
- divisor * 2t2 6t3 - 2t2
remainder 9t2 - 9t + 2
- divisor * 3t1 9t2 - 3t
remainder - 6t + 2
- divisor * -2t0 - 6t + 2
remainder 0
Quotient : 2t2+3t-2 Remainder: 0
Trying to factor by splitting the middle term
5.5 Factoring 2t2+3t-2
The first term is, 2t2 its coefficient is 2 .
The middle term is, +3t its coefficient is 3 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 2 • -2 = -4
Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is 3 .
-4 + 1 = -3
-2 + 2 = 0
-1 + 4 = 3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 4
2t2 - 1t + 4t - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
t • (2t-1)
Add up the last 2 terms, pulling out common factors :
2 • (2t-1)
Step-5 : Add up the four terms of step 4 :
(t+2) • (2t-1)
Which is the desired factorization
Trying to factor by splitting the middle term
5.6 Factoring 3t2+5t-2
The first term is, 3t2 its coefficient is 3 .
The middle term is, +5t its coefficient is 5 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 3 • -2 = -6
Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is 5 .
-6 + 1 = -5
-3 + 2 = -1
-2 + 3 = 1
-1 + 6 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 6
3t2 - 1t + 6t - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
t • (3t-1)
Add up the last 2 terms, pulling out common factors :
2 • (3t-1)
Step-5 : Add up the four terms of step 4 :
(t+2) • (3t-1)
Which is the desired factorization
Canceling Out :
5.7 Cancel out (t+2) which appears on both sides of the fraction line.
Canceling Out :
5.8 Cancel out (3t-1) which appears on both sides of the fraction line.
Final result :
(2t - 1) • (2t - 3)
