What is the quotient when 10x^3-3x^2-7x+3 is divided by 2x-1?

Step-by-step explanation:

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Equation at the end of step 1

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Equation at the end of step

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           10x3 - 3x2 - 7x + 3

Simplify   ———————————————————

                 2x - 1        

Checking for a perfect cube :

3.1    10x3 - 3x2 - 7x + 3  is not a perfect cube

Trying to factor by pulling out :

3.2      Factoring:  10x3 - 3x2 - 7x + 3  

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -7x + 3  

Group 2:  10x3 - 3x2  

Pull out from each group separately :

Group 1:   (-7x + 3) • (1) = (7x - 3) • (-1)

Group 2:   (10x - 3) • (x2)

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 :

3.3    Find roots (zeroes) of :       F(x) = 10x3 - 3x2 - 7x + 3

Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  10  and the Trailing Constant is  3.

The factor(s) are:

of the Leading Coefficient :  1,2 ,5 ,10

of the Trailing Constant :  1 ,3

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        -3.00      

     -1       2        -0.50        4.50      

     -1       5        -0.20        4.20      

     -1       10        -0.10        3.66      

     -3       1        -3.00        -273.00      

     -3       2        -1.50        -27.00      

     -3       5        -0.60        3.96      

     -3       10        -0.30        4.56      

     1       1        1.00        3.00      

     1       2        0.50        0.00      2x - 1  

     1       5        0.20        1.56      

     1       10        0.10        2.28      

     3       1        3.00        225.00      

     3       2        1.50        19.50      

     3       5        0.60        -0.12      

     3       10        0.30        0.90      

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

  10x3 - 3x2 - 7x + 3  

can be divided with  2x - 1  

Polynomial Long Division :

3.4    Polynomial Long Division

Dividing :  10x3 - 3x2 - 7x + 3  

                             ("Dividend")

By         :    2x - 1    ("Divisor")

dividend     10x3  -  3x2  -  7x  +  3  

- divisor  * 5x2     10x3  -  5x2          

remainder         2x2  -  7x  +  3  

- divisor  * x1         2x2  -  x      

remainder          -  6x  +  3  

- divisor  * -3x0          -  6x  +  3  

remainder                0

Quotient :  5x2+x-3  Remainder:  0  

Trying to factor by splitting the middle term

3.5     Factoring  5x2+x-3  

The first term is,  5x2  its coefficient is  5 .

The middle term is,  +x  its coefficient is  1 .

The last term, "the constant", is  -3  

Step-1 : Multiply the coefficient of the first term by the constant   5 • -3 = -15  

Step-2 : Find two factors of  -15  whose sum equals the coefficient of the middle term, which is   1 .

     -15    +    1    =    -14  

     -5    +    3    =    -2  

     -3    +    5    =    2  

     -1    +    15    =    14  

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Canceling Out :

3.6    Cancel out  (2x-1)  which appears on both sides of the fraction line.

Final result :

 5x2 + x - 3

idk im srry

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