Find the value of X (x3+10x2-27x+18=0)
Answers
Answer:
x3-10x2+27x-18=0
Three solutions were found :
x = 6
x = 3
x = 1
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((x3) - (2•5x2)) + 27x) - 18 = 0
Step 2 :
Checking for a perfect cube :
2.1 x3-10x2+27x-18 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-10x2+27x-18
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 27x-18
Group 2: x3-10x2
Pull out from each group separately :
Group 1: (3x-2) • (9)
Group 2: (x-10) • (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) = x3-10x2+27x-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 -56.00
-2 1 -2.00 -120.00
-3 1 -3.00 -216.00
-6 1 -6.00 -756.00
-9 1 -9.00 -1800.00
-18 1 -18.00 -9576.00
1 1 1.00 0.00 x-1
2 1 2.00 4.00
3 1 3.00 0.00 x-3
6 1 6.00 0.00 x-6
9 1 9.00 144.00
18 1 18.00 3060.00
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
x3-10x2+27x-18
can be divided by 3 different polynomials,including by x-6
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-10x2+27x-18
("Dividend")
By : x-6 ("Divisor")
dividend x3 - 10x2 + 27x - 18
- divisor * x2 x3 - 6x2
remainder - 4x2 + 27x - 18
- divisor * -4x1 - 4x2 + 24x
remainder 3x - 18
- divisor * 3x0 3x - 18
remainder 0
Quotient : x2-4x+3 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-4x+3
The first term is, x2 its coefficient is 1 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant 1 • 3 = 3
Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is -4 .
-3 + -1 = -4 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -3 and -1
x2 - 3x - 1x - 3
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-3)
Add up the last 2 terms, pulling out common factors :
1 • (x-3)
Step-5 : Add up the four terms of step 4 :
(x-1) • (x-3)
Which is the desired factorization