2x^3-x^2+3
——————
X-3

(x2 - x - 1) • (2x + 1)

Step-by-step explanation:

((2x3 - x2) - 3x) - 1

STEP

2

:

Checking for a perfect cube

2.1 2x3-x2-3x-1 is not a perfect cube

Trying to factor by pulling out :

2.2 Factoring: 2x3-x2-3x-1

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -3x-1

Group 2: 2x3-x2

Pull out from each group separately :

Group 1: (3x+1) • (-1)

Group 2: (2x-1) • (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 :

2.3 Find roots (zeroes) of : F(x) = 2x3-x2-3x-1

Polynomial 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 integers

The 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 Coefficient

In this case, the Leading Coefficient is 2 and the Trailing Constant is -1.

The factor(s) are:

of the Leading Coefficient : 1,2

of the Trailing Constant : 1

Let us test

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -1.00

-1 2 -0.50 0.00 2x+1

1 1 1.00 -3.00

1 2 0.50 -2.50

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-x2-3x-1

can be divided with 2x+1

Polynomial Long Division :

2.4 Polynomial Long Division

Dividing : 2x3-x2-3x-1

("Dividend")

By : 2x+1 ("Divisor")

dividend 2x3 - x2 - 3x - 1

- divisor * x2 2x3 + x2

remainder - 2x2 - 3x - 1

- divisor * -x1 - 2x2 - x

remainder - 2x - 1

- divisor * -x0 - 2x - 1

remainder 0

Quotient : x2-x-1 Remainder: 0

Trying to factor by splitting the middle term

2.5 Factoring x2-x-1

The first term is, x2 its coefficient is 1 .

The middle term is, -x its coefficient is -1 .

The last term, "the constant", is -1

Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1

Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is -1 .

-1 + 1 = 0

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Final result :

(x2 - x - 1) • (2x + 1)

the answer will be b x² +x-1

step-by-step explanation: