<span> 3.1 </span> <span> 2x3 - 4x2 - 3x - 9</span> is not a perfect cube
<span> 3.2 </span> Factoring: <span> 2x3 - 4x2 - 3x - 9</span>
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -3x - 9
Group 2: <span> 2x3 - 4x2</span>
Pull out from each group separately :
Group 1: (x + 3) • (-3)
Group 2: <span> (x - 2) • (2x2)</span>
<span>Bad news !! Factoring by pulling out fails :
</span>The groups have no common factor and can not be added up to form a multiplication.
<span> 3.3 </span> Find roots (zeroes) of : <span> F(x) = 2x3 - 4x2 - 3x - 9</span>
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 <span> -9.
</span>The factor(s) are:
of the Leading Coefficient : <span> 1,2
</span>of the Trailing Constant : <span> 1 ,3 ,9
</span>Let us test ....
The Factor Theorem states that if <span>P/Q </span>is root of a polynomial then this polynomial can be divided by <span>q*x-p </span>Note that <span>q and p originate from P/Q </span>reduced to its lowest terms
In our case this means that
<span> <span>2x3 - 4x2 - 3x - 9</span> </span>
can be divided with <span> x - 3 </span>
<span> 3.4 </span> Polynomial Long Division
Dividing : <span> <span>2x3 - 4x2 - 3x - 9</span>
("Dividend")
</span>By : <span> x - 3 ("Divisor")
</span>
Quotient : <span> <span>2x2+2x+3</span> </span>Remainder: <span> 0 </span>
Trying to factor by splitting the middle term<span> 3.5 </span> Factoring <span> 2x2+2x+3</span>
The first term is, <span> <span>2x2</span> </span> its coefficient is <span> 2 </span>.
The middle term is, <span> +2x </span> its coefficient is <span> 2 </span>.
The last term, "the constant", is <span> +3 </span>
Step-1 : Multiply the coefficient of the first term by the constant <span> <span> 2</span> • 3 = 6</span>
Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is <span> 2 </span>.
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
<span> 3.6 </span> Cancel out <span> (x-3) </span> which appears on both sides of the fraction line.
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