STEP 1:

Equation at the end of step 1

STEp 2:

4x3 - 3x + 9

Simplify ————————————

2x - 3

2.1 Find roots (zeroes) of : F(x) = 4x3 - 3x + 9

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 4 and the Trailing Constant is 9.

The factor(s) are:

of the Leading Coefficient : 1,2 ,4

of the Trailing Constant : 1 ,3 ,9

Let us test ....

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

-1 1 -1.00 8.00

-1 2 -0.50 10.00

-1 4 -0.25 9.69

-3 1 -3.00 -90.00

-3 2 -1.50 0.00 2x + 3

Note - For TIDINESS, printing of 13 checks which found no root was suppressed

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

4x3 - 3x + 9

can be divided with 2x + 3

Polynomial Long Division :

2.2 Polynomial Long Division

Dividing : 4x3 - 3x + 9

("Dividend")

By : 2x + 3 ("Divisor")

dividend 4x3 - 3x + 9

- divisor * 2x2 4x3 + 6x2

remainder - 6x2 - 3x + 9

- divisor * -3x1 - 6x2 - 9x

remainder 6x + 9

- divisor * 3x0 6x + 9

remainder 0

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

Trying to factor by splitting the MIDDLE term

2.3 Factoring 2x2-3x+3

The first term is, 2x2 its coefficient is 2 .

The middle term is, -3x its coefficient is -3 .

The last term, "the constant", is +3

Step-1 : Multiply the coefficient of the first term by the constant 2 • 3 = 6

Step-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

2.4 Polynomial Long Division

Dividing : 2x2-3x+3

("Dividend")

By : 2x-3 ("Divisor")

dividend 2x2 - 3x + 3

- divisor * x1 2x2 - 3x

remainder 3

- divisor * 0x0

remainder 3

Quotient : x

Remainder : 3

Final result :

(2x2 - 3x + 3) • (2x + 3)

———————————

2x - 3