(x3-3x2+3x-2)/(x2-x+1)
This deals with finding the roots (zeroes) of polynomials.
You are watching: What is the quotient (x3 – 3x2 + 3x – 2) ÷ (x2 – x + 1)?
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution: (1): "x2" was replaced by "x^2". 2 more similar replacement(s).Step 1 :
Equation at the end of step 1 :
Step 2 :
x3 - 3x2 + 3x - 2 Simplify ————————————————— x2 - x + 1 Checking for a perfect cube :2.1x3 - 3x2 + 3x - 2 is not a perfect cube Trying to factor by pulling out :2.2 Factoring: x3 - 3x2 + 3x - 2 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: 3x - 2Group 2: -3x2 + x3Pull out from each group separately :Group 1: (3x - 2) • (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 Find roots (zeroes) of : F(x) = x3 - 3x2 + 3x - 2Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 and the Trailing Constant is -2. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 Let us test ....
| -1 | 1 | -1.00 | -9.00 | ||||||
| -2 | 1 | -2.00 | -28.00 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 2 | 1 | 2.00 | 0.00 | x - 2 |
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 + 3x - 2can be divided with x - 2
Polynomial Long Division :
2.4 Polynomial Long Division Dividing : x3 - 3x2 + 3x - 2("Dividend") By:x - 2("Divisor")
| dividend | x3 | - | 3x2 | + | 3x | - | 2 | ||
| -divisor | * x2 | x3 | - | 2x2 | |||||
| remainder | - | x2 | + | 3x | - | 2 | |||
| -divisor | * -x1 | - | x2 | + | 2x | ||||
| remainder | x | - | 2 | ||||||
| -divisor | * x0 | x | - | 2 | |||||
| remainder | 0 |
Quotient : x2-x+1 Remainder: 0
Trying to factor by splitting the middle term2.5Factoring x2-x+1 The first term is, x2 its coefficient is 1.The middle term is, -x its coefficient is -1.The last term, "the constant", is +1Step-1 : Multiply the coefficient of the first term by the constant 1•1=1Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -1.
| -1 | + | -1 | = | -2 | ||
| 1 | + | 1 | = | 2 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored
Trying to factor by splitting the middle term2.6Factoring x2-x+1 The first term is, x2 its coefficient is 1.The middle term is, -x its coefficient is -1.The last term, "the constant", is +1Step-1 : Multiply the coefficient of the first term by the constant 1•1=1Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -1.
See more: Which Group On The Periodic Table Has Two Elements That Exist As Gases At Stp ?
| -1 | + | -1 | = | -2 | ||
| 1 | + | 1 | = | 2 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored
Canceling Out :2.7 Cancel out (x2-x+1) which appears on both sides of the fraction line.