Solve Polynomialrootcalculator 2x^3-3x^2-29x-30 Tiger Algebra ...

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Step 1 :

Equation at the end of step 1 :

(((2 • (x3)) - 3x2) - 29x) - 30

Step 2 :

Equation at the end of step 2 :

((2x3 - 3x2) - 29x) - 30

Step 3 :

Checking for a perfect cube :

3.1 2x3-3x2-29x-30 is not a perfect cube

Trying to factor by pulling out :

3.2 Factoring: 2x3-3x2-29x-30 Thoughtfully split the expression at hand into groups, each group having two terms :Group 1: -29x-30 Group 2: 2x3-3x2 Pull out from each group separately :Group 1: (29x+30) • (-1)Group 2: (2x-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 :

3.3 Find roots (zeroes) of : F(x) = 2x3-3x2-29x-30Polynomial Roots Calculator is a set of methods aimed at finding values of x for 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 number P/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 2 and the Trailing Constant is -30. The factor(s) are: of the Leading Coefficient : 1,2 of the Trailing Constant : 1 ,2 ,3 ,5 ,6 ,10 ,15 ,30 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 -6.00
-1 2 -0.50 -16.50
-2 1 -2.00 0.00 x+2
-3 1 -3.00 -24.00
-3 2 -1.50 0.00 2x+3
-5 1 -5.00 -210.00
-5 2 -2.50 -7.50
-6 1 -6.00 -396.00
-10 1 -10.00 -2040.00
-15 1 -15.00 -7020.00
-15 2 -7.50 -825.00
-30 1 -30.00 -55860.00
1 1 1.00 -60.00
1 2 0.50 -45.00
2 1 2.00 -84.00
3 1 3.00 -90.00
3 2 1.50 -73.50
5 1 5.00 0.00 x-5
5 2 2.50 -90.00
6 1 6.00 120.00
10 1 10.00 1380.00
15 1 15.00 5610.00
15 2 7.50 427.50
30 1 30.00 50400.00

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 2x3-3x2-29x-30 can be divided by 3 different polynomials,including by x-5

Polynomial Long Division :

3.4 Polynomial Long Division Dividing : 2x3-3x2-29x-30 ("Dividend") By : x-5 ("Divisor")

dividend 2x3 - 3x2 - 29x - 30
- divisor * 2x2 2x3 - 10x2
remainder 7x2 - 29x - 30
- divisor * 7x1 7x2 - 35x
remainder 6x - 30
- divisor * 6x0 6x - 30
remainder0

Quotient : 2x2+7x+6 Remainder: 0

Trying to factor by splitting the middle term

3.5 Factoring 2x2+7x+6 The first term is, 2x2 its coefficient is 2 .The middle term is, +7x its coefficient is 7 .The last term, "the constant", is +6 Step-1 : Multiply the coefficient of the first term by the constant 2 • 6 = 12 Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is 7 .

-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7
-3 + -4 = -7
-2 + -6 = -8
-1 + -12 = -13
1 + 12 = 13
2 + 6 = 8
3 + 4 = 7 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 4 2x2 + 3x + 4x + 6Step-4 : Add up the first 2 terms, pulling out like factors : x • (2x+3) Add up the last 2 terms, pulling out common factors : 2 • (2x+3) Step-5 : Add up the four terms of step 4 : (x+2) • (2x+3) Which is the desired factorization

Final result :

(2x + 3) • (x + 2) • (x - 5)

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