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Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
y-((3*x^6-7*x+9)/(7*x^2+7*x+9))=0
Step 1 :
Equation at the end of step 1 :
(((3•(x6))-7x)+9) y-————————————————— = 0 ((7x2+7x)+9)Step 2 :
Equation at the end of step 2 :
((3x6 - 7x) + 9)
y - ———————————————— = 0
(7x2 + 7x + 9)
Step 3 :
3x6 - 7x + 9
Simplify ————————————
7x2 + 7x + 9
Polynomial Roots Calculator :
3.1 Find roots (zeroes) of : F(x) = 3x6 - 7x + 9
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 9.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,3 ,9
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 19.00 | ||||||
| -1 | 3 | -0.33 | 11.34 | ||||||
| -3 | 1 | -3.00 | 2217.00 | ||||||
| -9 | 1 | -9.00 | 1594395.00 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 3 | 0.33 | 6.67 | ||||||
| 3 | 1 | 3.00 | 2175.00 | ||||||
| 9 | 1 | 9.00 | 1594269.00 |
Polynomial Roots Calculator found no rational roots
Trying to factor by splitting the middle term
3.2 Factoring 7x2 + 7x + 9
The first term is, 7x2 its coefficient is 7 .
The middle term is, +7x its coefficient is 7 .
The last term, "the constant", is +9
Step-1 : Multiply the coefficient of the first term by the constant 7 • 9 = 63
Step-2 : Find two factors of 63 whose sum equals the coefficient of the middle term, which is 7 .
| -63 | + | -1 | = | -64 | ||
| -21 | + | -3 | = | -24 | ||
| -9 | + | -7 | = | -16 | ||
| -7 | + | -9 | = | -16 | ||
| -3 | + | -21 | = | -24 | ||
| -1 | + | -63 | = | -64 | ||
| 1 | + | 63 | = | 64 | ||
| 3 | + | 21 | = | 24 | ||
| 7 | + | 9 | = | 16 | ||
| 9 | + | 7 | = | 16 | ||
| 21 | + | 3 | = | 24 | ||
| 63 | + | 1 | = | 64 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Polynomial Long Division :
3.3 Polynomial Long Division
Dividing : 3x6-7x+9
("Dividend")
By : 7x2+7x+9 ("Divisor")
| dividend | 3x6 | - | 7x | + | 9 | ||||||||||
| - divisor | * 0x4 | ||||||||||||||
| remainder | 3x6 | - | 7x | + | 9 | ||||||||||
| - divisor | * 0x3 | ||||||||||||||
| remainder | 3x6 | - | 7x | + | 9 | ||||||||||
| - divisor | * 0x2 | ||||||||||||||
| remainder | 3x6 | - | 7x | + | 9 | ||||||||||
| - divisor | * 0x1 | ||||||||||||||
| remainder | 3x6 | - | 7x | + | 9 | ||||||||||
| - divisor | * 0x0 | ||||||||||||||
| remainder | 3x6 | - | 7x | + | 9 |
Quotient : 0
Remainder : 3x6-7x+9
Equation at the end of step 3 :
(3x6 - 7x + 9)
y - —————————————— = 0
7x2 + 7x + 9
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using (7x2+7x+9) as the denominator :
y y • (7x2 + 7x + 9)
y = — = ——————————————————
1 (7x2 + 7x + 9)
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
4.2 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:
y • (7x2+7x+9) - ((3x6-7x+9)) 7yx2 + 7yx + 9y - 3x6 + 7x - 9
————————————————————————————— = ——————————————————————————————
1 • (7x2+7x+9) 1 • (7x2 + 7x + 9)
Trying to factor by pulling out :
4.3 Factoring: 7yx2 + 7yx + 9y - 3x6 + 7x - 9
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 9y - 9
Group 2: 7yx2 + 7yx
Group 3: -3x6 + 7x
Pull out from each group separately :
Group 1: (y - 1) • (9)
Group 2: (x + 1) • (7yx)
Group 3: (3x5 - 7) • (-x)
Looking for common sub-expressions :
Group 1: (y - 1) • (9)
Group 3: (3x5 - 7) • (-x)
Group 2: (x + 1) • (7yx)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Equation at the end of step 4 :
7yx2 + 7yx + 9y - 3x6 + 7x - 9
—————————————————————————————— = 0
7x2 + 7x + 9
Step 5 :
When a fraction equals zero :
5.1 When a fraction equals zero ...Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.
Here's how:
7yx2+7yx+9y-3x6+7x-9
———————————————————— • 7x2+7x+9 = 0 • 7x2+7x+9
(7x2+7x+9)
Now, on the left hand side, the 7x2+7x+9 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
7yx2+7yx+9y-3x6+7x-9 = 0
Solving a Single Variable Equation :
5.2 Solve 7yx2+7yx+9y3x6+7x-9 = 0
In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.
We shall not handle this type of equations at this time.
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