Question

Find the products :
(f) (9x^2-x+15) x (x^2-x-1)

Answer 1

+1/x2-3x=x+1/9x2-15x One solution was found :                         x ≓ -0.436203748

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2".  1 more similar replacement(s).

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 

                x+1/x^2-3*x-(x+1/9*x^2-15*x)=0 

Step by step solution :Step  1  : 1 Simplify — 9 Equation at the end of step  1  : 1 1 ((x+————)-3x)-((x+(—•x2))-15x) = 0 (x2) 9 Step  2  :Equation at the end of step  2  : 1 x2 ((x+————)-3x)-((x+——)-15x) = 0 (x2) 9 Step  3  :Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  9  as the denominator :

x x • 9 x = — = ————— 1 9

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • 9 + x2 x2 + 9x —————————— = ——————— 9 9 Equation at the end of step  3  : 1 (x2+9x) ((x+————)-3x)-(———————-15x) = 0 (x2) 9 Step  4  :Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  9  as the denominator :

15x 15x • 9 15x = ——— = ——————— 1 9 Step  5  :Pulling out like terms :

 5.1     Pull out like factors :

   x2 + 9x  =   x • (x + 9) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions 

x • (x+9) - (15x • 9) x2 - 126x ————————————————————— = ————————— 9 9 Equation at the end of step  5  : 1 (x2 - 126x) ((x + ————) - 3x) - ——————————— = 0 (x2) 9 Step  6  : 1 Simplify —— x2 Equation at the end of step  6  : 1 (x2 - 126x) ((x + ——) - 3x) - ——————————— = 0 x2 9 Step  7  :Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  x2  as the denominator :

x x • x2 x = — = —————— 1 x2 Adding fractions that have a common denominator :

 7.2       Adding up the two equivalent fractions 

x • x2 + 1 x3 + 1 —————————— = —————— x2 x2 Equation at the end of step  7  : (x3 + 1) (x2 - 126x) (———————— - 3x) - ——————————— = 0 x2 9 Step  8  :Rewriting the whole as an Equivalent Fraction :

 8.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x2  as the denominator :

3x 3x • x2 3x = —— = ——————— 1 x2 Trying to factor as a Sum of Cubes :

 8.2      Factoring:  x3 + 1 

Theory : A sum of two perfect cubes,  a3 + b3can be factored into  :
             (a+b) • (a2-ab+b2)
Proof  : (a+b) • (a2-ab+b2) = 
    a3-a2b+ab2+ba2-b2a+b3 =
    a3+(a2b-ba2)+(ab2-b2a)+b3=
    a3+0+0+b3=
    a3+b3

Check :  1  is the cube of   1 
Check :  x3 is the cube of   x1

Factorization is :
             (x + 1)  •  (x2 - x + 1) 

Trying to factor by splitting the middle term

 8.3     Factoring  x2 - x + 1 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  +1 

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

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

     -1   +   -1   =   -2     1   +   1   =   2


Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 8.4       Adding up the two equivalent fractions 

(x+1) • (x2-x+1) - (3x • x2) 1 - 2x3 ———————————————————————————— = ——————— x2 x2 Equation at the end of step  8  : (1 - 2x3) (x2 - 126x) ————————— - ——————————— = 0 x2 9 Step  9  :Trying to factor as a Difference of Cubes:

 9.1      Factoring:  1-2x3 

Theory : A difference of two perfect cubes,  a3 - b3 can be factored into
              (a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =
            a3+a2b+ab2-ba2-b2a-b3 =
            a3+(a2b-ba2)+(ab2-b2a)-b3 =
            a3+0+0+b3 =
            a3+b3

Check :  1  is the cube of  1 

Check :  2  is not a cube !! 
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 9.2    Find roots (zeroes) of :       F(x) = -2x3+1
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  -2. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      3.00        -2     1      -2.00      17.00        1     1      1.00      -1.00        2     1      2.00      -15.00   


Polynomial Roots Calculator found no

Answer 2

here's your answer...