# FINDING THE ROOTS OF POLYNOMIALS

### (x-1)(x-2)(x-3)(x-4)-24

This solution đơn hàng 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))-24

## Step 2 :

Equation at the over of step 2 : ((x-1)•(x-2)•(x-3)•(x-4))-24

## Step 3 :

Equation at the kết thúc of step 3 : (x-1)•(x-2)•(x-3)•(x-4)-24

## 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 và can not be added up to khung 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 set of methods aimed at finding values ofxfor which F(x)=0 Rational Roots thử nghiệm 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 kiểm tra ....

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PQP/QF(P/Q)Divisor
-11 -1.00 -96.00
-21 -2.00 -168.00
-51 -5.00 -600.00
-101-10.00-2400.00
-251-25.00-22800.00
-501-50.00-151800.00
11 1.00 -24.00
21 2.00 -12.00
51 5.00 0.00x - 5
101 10.00 300.00
251 25.0010200.00
501 50.00101700.00

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p lưu ý 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 factor by splitting the middle term

5.6Factoring 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