<span>tep 1 :</span> 3
Simplify —
x
<span>Equation at the end of step 1 :</span> 3
px - ((9x + —) + 4) = 0
x
<span>Step 2 :</span>Rewriting the whole as an Equivalent Fraction :
<span> 2.1 </span> Adding a fraction to a whole
Rewrite the whole as a fraction using <span> x </span> as the denominator :
9x 9x • x
9x = —— = ——————
1 x
<span>Equivalent fraction : </span>The fraction thus generated looks different but has the same value as the whole
<span>Common denominator : </span>The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
<span> 2.2 </span> Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
<span> 9x • x + 3 9x2 + 3
—————————— = ———————
x x
</span><span>Equation at the end of step 2 :</span><span> (9x2 + 3)
px - (————————— + 4) = 0
x
</span><span>Step 3 :</span>Rewriting the whole as an Equivalent Fraction :
<span> 3.1 </span> Adding a whole to a fraction
Rewrite the whole as a fraction using <span> x </span> as the denominator :
4 4 • x
4 = — = —————
1 x
<span>Step 4 :</span>Pulling out like terms :
<span> 4.1 </span> Pull out like factors :
<span> 9x2 + 3</span> = <span> 3 • (3x2 + 1)</span>
Polynomial Roots Calculator :
<span> 4.2 </span> Find roots (zeroes) of : <span> F(x) = 3x2 + 1</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 3 and the Trailing Constant is <span> 1.
</span>The factor(s) are:
of the Leading Coefficient : <span> 1,3
</span>of the Trailing Constant : <span> 1
</span>Let us test ....
<span><span> P Q P/Q F(P/Q) Divisor</span><span> -1 1 -1.00 4.00 </span><span> -1 3 -0.33 1.33 </span><span> 1 1 1.00 4.00 </span><span> 1 3 0.33 1.33 </span></span>
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
<span> 4.3 </span> Adding up the two equivalent fractions
<span> 3 • (3x2+1) + 4 • x 9x2 + 4x + 3
——————————————————— = ————————————
x x
</span><span>Equation at the end of step 4 :</span><span> (9x2 + 4x + 3)
px - —————————————— = 0
x
</span><span>Step 5 :</span>Rewriting the whole as an Equivalent Fraction :
<span> 5.1 </span> Subtracting a fraction from a whole
Rewrite the whole as a fraction using <span> x </span> as the denominator :
px px • x
px = —— = ——————
1 x
Trying to factor by splitting the middle term
<span> 5.2 </span> Factoring <span> 9x2 + 4x + 3</span>
The first term is, <span> <span>9x2</span> </span> its coefficient is <span> 9 </span>.
The middle term is, <span> +4x </span> its coefficient is <span> 4 </span>.
The last term, "the constant", is <span> +3 </span>
Step-1 : Multiply the coefficient of the first term by the constant <span> <span> 9</span> • 3 = 27</span>
Step-2 : Find two factors of 27 whose sum equals the coefficient of the middle term, which is <span> 4 </span>.
<span><span> -27 + -1 = -28</span><span> -9 + -3 = -12</span><span> -3 + -9 = -12</span><span> -1 + -27 = -28</span><span> 1 + 27 = 28</span><span> 3 + 9 = 12</span><span> 9 + 3 = 12</span><span> 27 + 1 = 28</span></span>
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
<span> 5.3 </span> Adding up the two equivalent fractions
<span> px • x - ((9x2+4x+3)) px2 - 9x2 - 4x - 3
————————————————————— = ——————————————————
x x
</span><span>Equation at the end of step 5 :</span><span> px2 - 9x2 - 4x - 3
—————————————————— = 0
x
</span><span>Step 6 :</span>When a fraction equals zero :<span><span> 6.1 </span> When a fraction equals zero ...</span>
Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the <span>denominator, </span>Tiger multiplys both sides of the equation by the denominator.
Here's how:
<span> px2-9x2-4x-3
———————————— • x = 0 • x
x
</span>
Now, on the left hand side, the <span> x </span> cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
<span> px2-9x2-4x-3</span> = 0
Solving a Single Variable Equation :
<span> 6.2 </span> Solve <span> <span>px2-9x2-4x-3</span> </span> = 0
In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.
