<span><span> (2x3-4x2-3x-9)/(x-3)</span> </span>Final result :<span> 2x2 + 2x + 3
</span>Step by step solution :<span>Step 1 :</span><span>Equation at the end of step 1 :</span><span>
<span> Step 2 :</span></span><span>Equation at the end of step 2 :</span>
<span>Step 3 :</span><span> 2x3 - 4x2 - 3x - 9
Simplify ——————————————————
x - 3
</span>Checking for a perfect cube :
<span> 3.1 </span> <span> 2x3 - 4x2 - 3x - 9</span> is not a perfect cube
Trying to factor by pulling out :
<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.
Polynomial Roots Calculator :
<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 ....
<span><span> P Q P/Q F(P/Q) Divisor</span><span> -1 1 -1.00 -12.00 </span><span> -1 2 -0.50 -8.75 </span><span> -3 1 -3.00 -90.00 </span><span> -3 2 -1.50 -20.25 </span><span> -9 1 -9.00 -1764.00 </span><span> -9 2 -4.50 -258.75 </span><span> 1 1 1.00 -14.00 </span><span> 1 2 0.50 -11.25 </span><span> 3 1 3.00 0.00 <span> x - 3 </span></span><span> 3 2 1.50 -15.75 </span><span> 9 1 9.00 1098.00 </span><span> 9 2 4.50 78.75 </span></span>
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>
Polynomial Long Division :
<span> 3.4 </span> Polynomial Long Division
Dividing : <span> <span>2x3 - 4x2 - 3x - 9</span>
("Dividend")
</span>By : <span> x - 3 ("Divisor")
</span>
<span><span>dividend <span> 2x3 </span>-<span> 4x2 </span>- 3x - 9 </span><span>- divisor<span> <span>* 2x2</span> </span> <span> 2x3 </span>-<span> 6x2 </span> </span><span>remainder <span> 2x2 </span>- 3x - 9 </span><span>- divisor<span> <span>* 2x1</span> </span> <span> 2x2 </span>- 6x </span><span>remainder 3x - 9 </span><span>- divisor<span> <span>* 3x0</span> </span> 3x - 9 </span><span>remainder 0</span></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>.
<span><span> -6 + -1 = -7</span><span> -3 + -2 = -5</span><span> -2 + -3 = -5</span><span> -1 + -6 = -7</span><span> 1 + 6 = 7</span><span> 2 + 3 = 5</span><span> 3 + 2 = 5</span><span> 6 + 1 = 7</span></span>
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Canceling Out :
<span> 3.6 </span> Cancel out <span> (x-3) </span> which appears on both sides of the fraction line.
Final result :<span> 2x2 + 2x + 3
</span><span>
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