FINDING THE ROOTS OF POLYNOMIALS
(x-1)(x-2)(x-3)(x-4)-24
This solution giao dịch with finding the roots (zeroes) of polynomials.
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Step by Step Solution
Step 1 :
Equation at the kết thúc of step 1 : (((x-1)•(x-2)•(x-3))•(x-4))-24Step 2 :
Equation at the over of step 2 : ((x-1)•(x-2)•(x-3)•(x-4))-24Step 3 :
Equation at the end of step 3 : (x-1)•(x-2)•(x-3)•(x-4)-24Step 4 :
Step 5 :
Pulling out like terms :5.1 Pull out like factors:x4 - 10x3 + 35x2 - 50x=x•(x3 - 10x2 + 35x - 50)Checking for a perfect cube :5.2x3 - 10x2 + 35x - 50 is not a perfect cube
Trying khổng lồ factor by pulling out :5.3 Factoring: x3 - 10x2 + 35x - 50 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: 35x - 50Group 2: -10x2 + x3Pull out from each group separately :Group 1: (7x - 10) • (5)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 size a multiplication.
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Polynomial Roots Calculator :
5.4 Find roots (zeroes) of : F(x) = x3 - 10x2 + 35x - 50Polynomial Roots Calculator is a phối of methods aimed at finding values ofxfor which F(x)=0 Rational Roots chạy thử 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 phường is a factor of the Trailing Constant & Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 và the Trailing Constant is -50. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,5 ,10 ,25 ,50 Let us test ....
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| -1 | 1 | -1.00 | -96.00 | ||||||
| -2 | 1 | -2.00 | -168.00 | ||||||
| -5 | 1 | -5.00 | -600.00 | ||||||
| -10 | 1 | -10.00 | -2400.00 | ||||||
| -25 | 1 | -25.00 | -22800.00 | ||||||
| -50 | 1 | -50.00 | -151800.00 | ||||||
| 1 | 1 | 1.00 | -24.00 | ||||||
| 2 | 1 | 2.00 | -12.00 | ||||||
| 5 | 1 | 5.00 | 0.00 | x - 5 | |||||
| 10 | 1 | 10.00 | 300.00 | ||||||
| 25 | 1 | 25.00 | 10200.00 | ||||||
| 50 | 1 | 50.00 | 101700.00 |
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 + 35x - 50can be divided with x - 5
Polynomial Long Division :
5.5 Polynomial Long Division Dividing : x3 - 10x2 + 35x - 50("Dividend") By:x - 5("Divisor")
| dividend | x3 | - | 10x2 | + | 35x | - | 50 | ||
| -divisor | * x2 | x3 | - | 5x2 | |||||
| remainder | - | 5x2 | + | 35x | - | 50 | |||
| -divisor | * -5x1 | - | 5x2 | + | 25x | ||||
| remainder | 10x | - | 50 | ||||||
| -divisor | * 10x0 | 10x | - | 50 | |||||
| remainder | 0 |
Quotient : x2-5x+10 Remainder: 0
Trying to lớn factor by splitting the middle term5.6Factoring x2-5x+10 The first term is, x2 its coefficient is 1.The middle term is, -5x its coefficient is -5.The last term, "the constant", is +10Step-1 : Multiply the coefficient of the first term by the constant 1•10=10Step-2 : Find two factors of 10 whose sum equals the coefficient of the middle term, which is -5.
| -10 | + | -1 | = | -11 | ||
| -5 | + | -2 | = | -7 | ||
| -2 | + | -5 | = | -7 | ||
| -1 | + | -10 | = | -11 | ||
| 1 | + | 10 | = | 11 | ||
| 2 | + | 5 | = | 7 | ||
| 5 | + | 2 | = | 7 | ||
| 10 | + | 1 | = | 11 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored
