3.2t + 7(–1.7t − 0.72) = –0.6t + 0.46 − 7.1t

what is t?

 (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)

S = (19+23)/2×14

S = 42/2×14

S = 21×14

S = 294 cm^2