Math, asked by arora1samarth, 8 months ago

Factorise 3x^3 - 23x^2 + 44x - 20 using factor theorem
Pls helppp

Answers

Answered by siddhantprasad8
1

Step  1  :

           20

Simplify   ——

           x  

Equation at the end of step  1  :

                             20

 ((((3•(x3))-(23•(x2)))+44x)-——)-5

                             x  

Step  2  :

Equation at the end of step  2  :

                        20

 ((((3•(x3))-23x2)+44x)-——)-5

                        x  

Step  3  :

Equation at the end of step  3  :

                            20      

 (((3x3 -  23x2) +  44x) -  ——) -  5

                            x      

Step  4  :

Rewriting the whole as an Equivalent Fraction :

4.1   Subtracting a fraction from a whole

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

                        3x3 - 23x2 + 44x     (3x3 - 23x2 + 44x) • x

    3x3 - 23x2 + 44x =  ————————————————  =  ——————————————————————

                               1                       x            

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

Step  5  :

Pulling out like terms :

5.1     Pull out like factors :

  3x3 - 23x2 + 44x  =   x • (3x2 - 23x + 44)  

Trying to factor by splitting the middle term

5.2     Factoring  3x2 - 23x + 44  

The first term is,  3x2  its coefficient is  3 .

The middle term is,  -23x  its coefficient is  -23 .

The last term, "the constant", is  +44  

Step-1 : Multiply the coefficient of the first term by the constant   3 • 44 = 132  

Step-2 : Find two factors of  132  whose sum equals the coefficient of the middle term, which is   -23 .

     -132    +    -1    =    -133  

     -66    +    -2    =    -68  

     -44    +    -3    =    -47  

     -33    +    -4    =    -37  

     -22    +    -6    =    -28  

     -12    +    -11    =    -23    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -12  and  -11  

                    3x2 - 12x - 11x - 44

Step-4 : Add up the first 2 terms, pulling out like factors :

                   3x • (x-4)

             Add up the last 2 terms, pulling out common factors :

                   11 • (x-4)

Step-5 : Add up the four terms of step 4 :

                   (3x-11)  •  (x-4)

            Which is the desired factorization

Adding fractions that have a common denominator :

5.3       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 • (x-4) • (3x-11) • x - (20)     3x4 - 23x3 + 44x2 - 20  

——————————————————————————————  =  ——————————————————————

              x                              x            

Equation at the end of step  5  :

 (3x4 - 23x3 + 44x2 - 20)      

 ———————————————————————— -  5

            x                

Step  6  :

Rewriting the whole as an Equivalent Fraction :

6.1   Subtracting a whole from a fraction

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

        5     5 • x

   5 =  —  =  —————

        1       x  

Checking for a perfect cube :

6.2    3x4 - 23x3 + 44x2 - 20  is not a perfect cube

Trying to factor by pulling out :

6.3      Factoring:  3x4 - 23x3 + 44x2 - 20  

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

Group 1:  44x2 - 20  

Group 2:  3x4 - 23x3  

Pull out from each group separately :

Group 1:   (11x2 - 5) • (4)

Group 2:   (3x - 23) • (x3)

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 :

6.4    Find roots (zeroes) of :       F(x) = 3x4 - 23x3 + 44x2 - 20

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  3  and the Trailing Constant is  -20.

The factor(s) are:

of the Leading Coefficient :  1,3

of the Trailing Constant :  1 ,2 ,4 ,5 ,10 ,20

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        50.00      

     -1       3        -0.33        -14.22      

     -2       1        -2.00        388.00      

     -2       3        -0.67        6.96      

     -4       1        -4.00        2924.00      

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

6.5       Adding up the two equivalent fractions

(3x4-23x3+44x2-20) - (5 • x)      3x4 - 23x3 + 44x2 - 5x - 20  

————————————————————————————  =  ———————————————————————————

             x                                x              

Polynomial Roots Calculator :

6.6    Find roots (zeroes) of :       F(x) = 3x4 - 23x3 + 44x2 - 5x - 20

    See theory in step 6.4

In this case, the Leading Coefficient is  3  and the Trailing Constant is  -20.

The factor(s) are:

of the Leading Coefficient :  1,3

of the Trailing Constant :  1 ,2 ,4 ,5 ,10 ,20

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        55.00      

     -1       3        -0.33        -12.56      

     -2       1        -2.00        398.00      

     -2       3        -0.67        10.30      

     -4       1        -4.00        2944.00      

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

 3x4 - 23x3 + 44x2 - 5x - 20  

 ———————————————————————————

              x            

Answered by Anonymous
2

Answer:

3x2-75

Step 1 : Equation at the end of step 1 : 3x2 - 75.

Step 2 :

Step 3 : Pulling out like terms : 3.1 Pull out like factors : 3x2 - 75 = 3 • (x2 - 25) Trying to factor as a Difference of Squares : 3.2 Factoring: x2 - 25. Check : 25 is the square of 5. Check : x2 is the square of x1 Factorization is : (x + 5) • (x - 5)

Step-by-step explanation:

inbthis way you can solve that question

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