Step 1 :
Equation at the kết thúc of step 1 : ((x • (x - 1) • (x - 2)) • (x - 3)) - 24Step 2 :
Equation at the over of step 2 : (x • (x - 1) • (x - 2) • (x - 3)) - 24Step 3 :
Equation at the kết thúc of step 3 : x • (x - 1) • (x - 2) • (x - 3) - 24Step 4 :
Polynomial Roots Calculator :
4.1 Find roots (zeroes) of : F(x) = x4-6x3+11x2-6x-24Polynomial Roots Calculator is a phối 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 phường is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 và the Trailing Constant is -24. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24 Let us kiểm tra ....Bạn đang xem: The practical design of steel
| -1 | 1 | -1.00 | 0.00 | x+1 | |||||
| -2 | 1 | -2.00 | 96.00 | ||||||
| -3 | 1 | -3.00 | 336.00 | ||||||
| -4 | 1 | -4.00 | 816.00 | ||||||
| -6 | 1 | -6.00 | 3000.00 | ||||||
| -8 | 1 | -8.00 | 7896.00 | ||||||
| -12 | 1 | -12.00 | 32736.00 | ||||||
| -24 | 1 | -24.00 | 421176.00 | ||||||
| 1 | 1 | 1.00 | -24.00 | ||||||
| 2 | 1 | 2.00 | -24.00 | ||||||
| 3 | 1 | 3.00 | -24.00 | ||||||
| 4 | 1 | 4.00 | 0.00 | x-4 | |||||
| 6 | 1 | 6.00 | 336.00 | ||||||
| 8 | 1 | 8.00 | 1656.00 | ||||||
| 12 | 1 | 12.00 | 11856.00 | ||||||
| 24 | 1 | 24.00 | 255000.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p cảnh báo that q and p originate from P/Q reduced to lớn its lowest terms In our case this means that x4-6x3+11x2-6x-24can be divided by 2 different polynomials,including by x-4
Polynomial Long Division :
4.2 Polynomial Long Division Dividing : x4-6x3+11x2-6x-24("Dividend") By:x-4("Divisor")
| dividend | x4 | - | 6x3 | + | 11x2 | - | 6x | - | 24 | ||
| -divisor | * x3 | x4 | - | 4x3 | |||||||
| remainder | - | 2x3 | + | 11x2 | - | 6x | - | 24 | |||
| -divisor | * -2x2 | - | 2x3 | + | 8x2 | ||||||
| remainder | 3x2 | - | 6x | - | 24 | ||||||
| -divisor | * 3x1 | 3x2 | - | 12x | |||||||
| remainder | 6x | - | 24 | ||||||||
| -divisor | * 6x0 | 6x | - | 24 | |||||||
| remainder | 0 |
Quotient : x3-2x2+3x+6 Remainder: 0
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = x3-2x2+3x+6See theory in step 4.1 In this case, the Leading Coefficient is 1 & the Trailing Constant is 6.
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The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,3 ,6 Let us thử nghiệm ....
| -1 | 1 | -1.00 | 0.00 | x+1 | |||||
| -2 | 1 | -2.00 | -16.00 | ||||||
| -3 | 1 | -3.00 | -48.00 | ||||||
| -6 | 1 | -6.00 | -300.00 | ||||||
| 1 | 1 | 1.00 | 8.00 | ||||||
| 2 | 1 | 2.00 | 12.00 | ||||||
| 3 | 1 | 3.00 | 24.00 | ||||||
| 6 | 1 | 6.00 | 168.00 |
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 lớn its lowest terms In our case this means that x3-2x2+3x+6can be divided with x+1
Polynomial Long Division :
4.4 Polynomial Long Division Dividing : x3-2x2+3x+6("Dividend") By:x+1("Divisor")
| dividend | x3 | - | 2x2 | + | 3x | + | 6 | ||
| -divisor | * x2 | x3 | + | x2 | |||||
| remainder | - | 3x2 | + | 3x | + | 6 | |||
| -divisor | * -3x1 | - | 3x2 | - | 3x | ||||
| remainder | 6x | + | 6 | ||||||
| -divisor | * 6x0 | 6x | + | 6 | |||||
| remainder | 0 |
Quotient : x2-3x+6 Remainder: 0
Trying to lớn factor by splitting the middle term4.5Factoring x2-3x+6 The first term is, x2 its coefficient is 1.The middle term is, -3x its coefficient is -3.The last term, "the constant", is +6Step-1 : Multiply the coefficient of the first term by the constant 1•6=6Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is -3.
| -6 | + | -1 | = | -7 | ||
| -3 | + | -2 | = | -5 | ||
| -2 | + | -3 | = | -5 | ||
| -1 | + | -6 | = | -7 | ||
| 1 | + | 6 | = | 7 | ||
| 2 | + | 3 | = | 5 | ||
| 3 | + | 2 | = | 5 | ||
| 6 | + | 1 | = | 7 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored