Question

Find the Value of K if the Polynomial p(x) = kx cube + 9x sq. +4x -10 is divided by (x-3) leaves a remainder -22

Answer 1

When ( kx³ + 9x² + 4x - 10 ) is divided by ( x - 3 ), it leaves - 22 as remainder.

If we subtract - 22 from the given polynomial, then the whole polynomial will be divisible by ( x - 3 ).

•°•

The polynomial formed -

= kx³ + 9x² + 4x - 10 - ( - 22 )

= kx³ + 9x² + 4x - 10 + 22

= kx³ + 9x² + 4x - 12

Now,

( x - 3 ) is a factor of polynomial.

So, it can make the whole polynomial equals to 0.

Hence,

x - 3 = 0

x = 3

Also, p ( 3 ) = 0

Put this value in the polynomial, we get

p ( 3 ) = k ( 3 )³ + 9 ( 3 )² + 4 ( 3 ) - 12

0 = 27 k + 81 + 12 - 12

0 = 27 k + 81

- 27 k = 81

k = - 3

Answer 2

x4-18x2+32x-15=0 

Three solutions were found :

 x = 3

 x = 1

 x = -5

Step by step solution :

Step  1  :

Equation at the end of step  1  :

(((x4) - (2•32x2)) + 32x) - 15 = 0

Step  2  :

Checking for a perfect cube :

 2.1    x4-18x2+32x-15  is not a perfect cube 

Trying to factor by pulling out :

 2.2      Factoring:  x4-18x2+32x-15 

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

Group 1:  32x-15 
Group 2:  -18x2+x4 

Pull out from each group separately :

Group 1:   (32x-15) • (1)
Group 2:   (x2-18) • (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) = x4-18x2+32x-15
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  -15. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,5 ,15 

 Let us test ....
  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      -64.00        -3     1      -3.00      -192.00        -5     1      -5.00      0.00    x+5      -15     1     -15.00     46080.00        1     1      1.00      0.00    x-1      3     1      3.00      0.00    x-3      5     1      5.00      320.00        15     1      15.00     47040.00   

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 
   x4-18x2+32x-15 
can be divided by 3 different polynomials,including by  x-3 

Polynomial Long Division :

 2.4    Polynomial Long Division 
Dividing :  x4-18x2+32x-15 
                              ("Dividend")
By         :    x-3    ("Divisor")
dividend  x4   - 18x2 + 32x - 15 - divisor * x3   x4 - 3x3       remainder    3x3 - 18x2 + 32x - 15 - divisor * 3x2     3x3 - 9x2     remainder    - 9x2 + 32x - 15 - divisor * -9x1     - 9x2 + 27x   remainder        5x - 15 - divisor * 5x0         5x - 15 remainder         0
Quotient :  x3+3x2-9x+5  Remainder:  0 

Polynomial Roots Calculator :

 2.5    Find roots (zeroes) of :       F(x) = x3+3x2-9x+5

     See theory in step 2.3 
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      0.00    x+5      1     1      1.00      0.00    x-1      5     1      5.00      160.00   

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+3x2-9x+5 
can be divided by 2 different polynomials,including by  x-1 

Polynomial Long Division :

 2.6    Polynomial Long Division 
Dividing :  x3+3x2-9x+5 
                              ("Dividend")
By         :    x-1    ("Divisor")
dividend  x3 + 3x2 - 9x + 5 - divisor * x2   x3 - x2     remainder    4x2 - 9x + 5 - divisor * 4x1     4x2 - 4x   remainder    - 5x + 5 - divisor * -5x0     - 5x + 5 remainder       0
Quotient :  x2+4x-5  Remainder:  0 

Trying to factor by splitting the middle term

 2.7     Factoring  x2+4x-5 

The first term is,  x2  its coefficient is  1 .
The middle term is,  +4x  its coefficient is  4 .
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   4 .
     -5   +   1   =   -4     -1   +   5   =   4   That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  5 
                     x2 - 1x + 5x - 5

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-1)
              Add up the last 2 terms, pulling out common factors :
                    5 • (x-1)
Step-5 : Add up the four terms of step 4 :
                    (x+5)  •  (x-1)
             Which is the desired factorization

Multiplying Exponential Expressions :

 2.8    Multiply  (x-1)  by  (x-1) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-1)  and the exponents are :
          1 , as  (x-1)  is the same number as  (x-1)1 
 and   1 , as  (x-1)  is the same number as  (x-1)1 
The product is therefore,  (x-1)(1+1) = (x-1)2