Solve Polynomialrootcalculator (2x^3-9x^2-x+30)/(x ... - Tiger Algebra

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Step 1 :

Equation at the end of step 1 :

Step 2 :

Equation at the end of step 2 :

Step 3 :

2x3 - 9x2 - x + 30 Simplify —————————————————— x - 3

Checking for a perfect cube :

3.1 2x3 - 9x2 - x + 30 is not a perfect cube

Trying to factor by pulling out :

3.2 Factoring: 2x3 - 9x2 - x + 30 Thoughtfully split the expression at hand into groups, each group having two terms :Group 1: -x + 30 Group 2: 2x3 - 9x2 Pull out from each group separately :Group 1: (-x + 30) • (1) = (x - 30) • (-1)Group 2: (2x - 9) • (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 - 9x2 - x + 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 20.00
-1 2 -0.50 28.00
-2 1 -2.00 -20.00
-3 1 -3.00 -102.00
-3 2 -1.50 4.50
-5 1 -5.00 -440.00
-5 2 -2.50 -55.00
-6 1 -6.00 -720.00
-10 1 -10.00 -2860.00
-15 1 -15.00 -8730.00
-15 2 -7.50 -1312.50
-30 1 -30.00 -62040.00
1 1 1.00 22.00
1 2 0.50 27.50
2 1 2.00 8.00
3 1 3.00 0.00 x - 3
3 2 1.50 15.00
5 1 5.00 50.00
5 2 2.50 2.50
6 1 6.00 132.00
10 1 10.00 1120.00
15 1 15.00 4740.00
15 2 7.50 360.00
30 1 30.00 45900.00

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 - 9x2 - x + 30 can be divided with x - 3

Polynomial Long Division :

3.4 Polynomial Long Division Dividing : 2x3 - 9x2 - x + 30 ("Dividend") By : x - 3 ("Divisor")

dividend 2x3 - 9x2 - x + 30
- divisor * 2x2 2x3 - 6x2
remainder- 3x2 - x + 30
- divisor * -3x1 - 3x2 + 9x
remainder- 10x + 30
- divisor * -10x0 - 10x + 30
remainder0

Quotient : 2x2-3x-10 Remainder: 0

Trying to factor by splitting the middle term

3.5 Factoring 2x2-3x-10 The first term is, 2x2 its coefficient is 2 .The middle term is, -3x its coefficient is -3 .The last term, "the constant", is -10 Step-1 : Multiply the coefficient of the first term by the constant 2 • -10 = -20 Step-2 : Find two factors of -20 whose sum equals the coefficient of the middle term, which is -3 .

-20 + 1 = -19
-10 + 2 = -8
-5 + 4 = -1
-4 + 5 = 1
-2 + 10 = 8
-1 + 20 = 19

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Canceling Out :

3.6 Cancel out (x-3) which appears on both sides of the fraction line.

Final result :

2x2 - 3x - 10

Từ khóa » F(x)=x^3-9x G(x)=x^2-2x-3