x3+5x2-18
Final result :
(x2 + 2x - 6) • (x + 3)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((x3) + 5x2) - 18
Step 2 :
Polynomial Roots Calculator :
2.1 Find roots (zeroes) of : F(x) = x3+5x2-18
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 -18.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6 ,9 ,18
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -14.00
-2 1 -2.00 -6.00
-3 1 -3.00 0.00 x+3
-6 1 -6.00 -54.00
-9 1 -9.00 -342.00
-18 1 -18.00 -4230.00
1 1 1.00 -12.00
2 1 2.00 10.00
3 1 3.00 54.00
6 1 6.00 378.00
9 1 9.00 1116.00
18 1 18.00 7434.00
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+5x2-18
can be divided with x+3
Polynomial Long Division :
2.2 Polynomial Long Division
Dividing : x3+5x2-18
("Dividend")
By : x+3 ("Divisor")
dividend x3 + 5x2 - 18
- divisor * x2 x3 + 3x2
remainder 2x2 - 18
- divisor * 2x1 2x2 + 6x
remainder - 6x - 18
- divisor * -6x0 - 6x - 18
remainder 0
Quotient : x2+2x-6 Remainder: 0
Trying to factor by splitting the middle term
2.3 Factoring x2+2x-6
The first term is, x2 its coefficient is 1 .
The middle term is, +2x its coefficient is 2 .
The last term, "the constant", is -6
Step-1 : Multiply the coefficient of the first term by the constant 1 • -6 = -6
Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is 2 .
-6 + 1 = -5
-3 + 2 = -1
-2 + 3 = 1
-1 + 6 = 5
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(x2 + 2x - 6) • (x + 3)
