## Step 1 :

Equation at the kết thúc of step 1 : ((x • (x - 1) • (x - 2)) • (x - 3)) - 24

## Step 2 :

Equation at the over of step 2 : (x • (x - 1) • (x - 2) • (x - 3)) - 24

## Step 3 :

Equation at the kết thúc of step 3 : x • (x - 1) • (x - 2) • (x - 3) - 24

## Step 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 ....

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PQP/QF(P/Q)Divisor
-11 -1.00 0.00x+1
-21 -2.00 96.00
-31 -3.00 336.00
-41 -4.00 816.00
-61 -6.00 3000.00
-81 -8.00 7896.00
-121-12.0032736.00
-241-24.00421176.00
11 1.00 -24.00
21 2.00 -24.00
31 3.00 -24.00
41 4.00 0.00x-4
61 6.00 336.00
81 8.00 1656.00
121 12.0011856.00
241 24.00255000.00

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 ....

PQP/QF(P/Q)Divisor
-11 -1.00 0.00x+1
-21 -2.00 -16.00
-31 -3.00 -48.00
-61 -6.00 -300.00
11 1.00 8.00
21 2.00 12.00
31 3.00 24.00
61 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 term

4.5Factoring 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