Question

p(x) = x3 – 3x2 + 7x – 5​

Answer 1

Answer:

Simplify ————————————————— x - 1

Checking for a perfect cube :

 2.1    x3 - 3x2 + 7x - 5  is not a perfect cube

Trying to factor by pulling out :

 2.2      Factoring:  x3 - 3x2 + 7x - 5 

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

Group 1:  7x - 5 

Group 2:  x3 - 3x2 

Pull out from each group separately :

Group 1:   (7x - 5) • (1)

Group 2:   (x - 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 :

 2.3    Find roots (zeroes) of :       F(x) = x3 - 3x2 + 7x - 5

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  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      -16.00        -5     1      -5.00      -240.00        1     1      1.00      0.00    x - 1      5     1      5.00      80.00   

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

   x3 - 3x2 + 7x - 5 

can be divided with  x - 1 

Polynomial Long Division :

 2.4    Polynomial Long Division

Dividing :  x3 - 3x2 + 7x - 5 

                              ("Dividend")

By         :    x - 1    ("Divisor")

dividend  x3 - 3x2 + 7x - 5 - divisor * x2   x3 - x2     remainder  - 2x2 + 7x - 5 - divisor * -2x1   - 2x2 + 2x   remainder      5x - 5 - divisor * 5x0       5x - 5 remainder       0

Quotient :  x2-2x+5  Remainder:  0 

Trying to factor by splitting the middle term

 2.5     Factoring  x2-2x+5 

The first term is,  x2  its coefficient is  1 .

The middle term is,  -2x  its coefficient is  -2 .

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   -2 .

     -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

Canceling Out :

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

Final result :

x2 - 2x + 5