Step-by-step explanation:
1 result(s) found
x Â
2
−x−3
Step by Step Solution:
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STEP
1
:
Equation at the end of step 1
Â
STEP
2
:
      x3 + 3x2 - 7x - 12
Simplify  ——————————————————
         x + 4    Â
Checking for a perfect cube :
2.1 Â Â x3 + 3x2 - 7x - 12 Â is not a perfect cube
Trying to factor by pulling out :
2.2 Â Â Â Factoring: Â x3 + 3x2 - 7x - 12 Â
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: Â -7x - 12 Â
Group 2: Â x3 + 3x2 Â
Pull out from each group separately :
Group 1:  (7x + 12) • (-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 Â Â Find roots (zeroes) of : Â Â Â F(x) = x3 + 3x2 - 7x - 12
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which  F(x)=0 Â
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  -12.
The factor(s) are:
of the Leading Coefficient : Â 1
of the Trailing Constant : Â 1 ,2 ,3 ,4 ,6 ,12
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -3.00   Â
   -2    1     -2.00     6.00   Â
   -3    1     -3.00     9.00   Â
   -4    1     -4.00     0.00    x + 4 Â
   -6    1     -6.00     -78.00   Â
   -12    1    -12.00    -1224.00   Â
   1    1     1.00     -15.00   Â
   2    1     2.00     -6.00   Â
   3    1     3.00     21.00   Â
   4    1     4.00     72.00   Â
   6    1     6.00     270.00   Â
   12    1     12.00     2064.00   Â
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 - 12 Â
can be divided with  x + 4 Â
Polynomial Long Division :
2.4 Â Â Polynomial Long Division
Dividing : Â x3 + 3x2 - 7x - 12 Â
               ("Dividend")
By     :   x + 4   ("Divisor")
dividend   x3  +  3x2  -  7x  -  12 Â
- divisor  * x2   x3  +  4x2     Â
remainder    -  x2  -  7x  -  12 Â
- divisor  * -x1    -  x2  -  4x   Â
remainder      -  3x  -  12 Â
- divisor  * -3x0      -  3x  -  12 Â
remainder         0
Quotient : Â x2-x-3 Â Remainder: Â 0 Â
Trying to factor by splitting the middle term
2.5   Factoring  x2-x-3 Â
The first term is,  x2  its coefficient is  1 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  -3 Â
Step-1 : Multiply the coefficient of the first term by the constant  1 • -3 = -3 Â
Step-2 : Find two factors of  -3  whose sum equals the coefficient of the middle term, which is  -1 .
   -3   +   1   =   -2 Â
   -1   +   3   =   2 Â
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Canceling Out :
2.6   Cancel out  (x+4)  which appears on both sides of the fraction line.
Final result :
 x2 - x - 3
Terms and topics
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Polynomial root calculator
Polynomial long division
Related links
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Root Finder -- Polynomials Calculator
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Polynomial long division - Wikipedia
Polynomial Long Division
Polynomials - Long Division