Solution - Finding the roots of polynomials
Other Ways to Solve
Finding the roots of polynomialsStep by Step Solution
Step 1 :
Equation at the end of step 1 :
k ((((6•(k3))-(3•(k2)))-((3•—)•(k3)))-23k2)+4k 4Step 2 :
k Simplify — 4
Equation at the end of step 2 :
k
((((6•(k3))-(3•(k2)))-((3•—)•k3))-23k2)+4k
4
Step 3 :
Multiplying exponential expressions :
3.1 k1 multiplied by k3 = k(1 + 3) = k4
Equation at the end of step 3 :
3k4 ((((6•(k3))-(3•(k2)))-———)-23k2)+4k 4Step 4 :
Equation at the end of step 4 :
3k4 ((((6•(k3))-3k2)-———)-23k2)+4k 4Step 5 :
Equation at the end of step 5 :
3k4
((((2•3k3) - 3k2) - ———) - 23k2) + 4k
4
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 4 as the denominator :
6k3 - 3k2 (6k3 - 3k2) • 4
6k3 - 3k2 = ————————— = ———————————————
1 4
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
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
6k3 - 3k2 = 3k2 • (2k - 1)
Adding fractions that have a common denominator :
7.2 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:
3k2 • (2k-1) • 4 - (3k4) -3k4 + 24k3 - 12k2
———————————————————————— = ——————————————————
4 4
Equation at the end of step 7 :
(-3k4 + 24k3 - 12k2)
(———————————————————— - 23k2) + 4k
4
Step 8 :
Rewriting the whole as an Equivalent Fraction :
8.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 4 as the denominator :
23k2 23k2 • 4
23k2 = ———— = ————————
1 4
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
-3k4 + 24k3 - 12k2 = -3k2 • (k2 - 8k + 4)
Trying to factor by splitting the middle term
9.2 Factoring k2 - 8k + 4
The first term is, k2 its coefficient is 1 .
The middle term is, -8k its coefficient is -8 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -8 .
| -4 | + | -1 | = | -5 | ||
| -2 | + | -2 | = | -4 | ||
| -1 | + | -4 | = | -5 | ||
| 1 | + | 4 | = | 5 | ||
| 2 | + | 2 | = | 4 | ||
| 4 | + | 1 | = | 5 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
9.3 Adding up the two equivalent fractions
-3k2 • (k2-8k+4) - (23k2 • 4) -3k4 + 24k3 - 44k2
————————————————————————————— = ——————————————————
4 4
Equation at the end of step 9 :
(-3k4 + 24k3 - 44k2)
———————————————————— + 4k
4
Step 10 :
Rewriting the whole as an Equivalent Fraction :
10.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 4 as the denominator :
4k 4k • 4
4k = —— = ——————
1 4
Step 11 :
Pulling out like terms :
11.1 Pull out like factors :
-3k4 + 24k3 - 44k2 = -k2 • (3k2 - 24k + 44)
Trying to factor by splitting the middle term
11.2 Factoring 3k2 - 24k + 44
The first term is, 3k2 its coefficient is 3 .
The middle term is, -24k its coefficient is -24 .
The last term, "the constant", is +44
Step-1 : Multiply the coefficient of the first term by the constant 3 • 44 = 132
Step-2 : Find two factors of 132 whose sum equals the coefficient of the middle term, which is -24 .
| -132 | + | -1 | = | -133 | ||
| -66 | + | -2 | = | -68 | ||
| -44 | + | -3 | = | -47 | ||
| -33 | + | -4 | = | -37 | ||
| -22 | + | -6 | = | -28 | ||
| -12 | + | -11 | = | -23 |
For tidiness, printing of 18 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
11.3 Adding up the two equivalent fractions
-k2 • (3k2-24k+44) + 4k • 4 -3k4 + 24k3 - 44k2 + 16k
——————————————————————————— = ————————————————————————
4 4
Step 12 :
Pulling out like terms :
12.1 Pull out like factors :
-3k4 + 24k3 - 44k2 + 16k =
-k • (3k3 - 24k2 + 44k - 16)
Checking for a perfect cube :
12.2 3k3 - 24k2 + 44k - 16 is not a perfect cube
Trying to factor by pulling out :
12.3 Factoring: 3k3 - 24k2 + 44k - 16
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 44k - 16
Group 2: -24k2 + 3k3
Pull out from each group separately :
Group 1: (11k - 4) • (4)
Group 2: (k - 8) • (3k2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
12.4 Find roots (zeroes) of : F(k) = 3k3 - 24k2 + 44k - 16
Polynomial Roots Calculator is a set of methods aimed at finding values of k for which F(k)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers k 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 -16.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4 ,8 ,16
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -87.00 | ||||||
| -1 | 3 | -0.33 | -33.44 | ||||||
| -2 | 1 | -2.00 | -224.00 | ||||||
| -2 | 3 | -0.67 | -56.89 | ||||||
| -4 | 1 | -4.00 | -768.00 | ||||||
| -4 | 3 | -1.33 | -124.44 | ||||||
| -8 | 1 | -8.00 | -3440.00 | ||||||
| -8 | 3 | -2.67 | -360.89 | ||||||
| -16 | 1 | -16.00 | -19152.00 | ||||||
| -16 | 3 | -5.33 | -1388.44 | ||||||
| 1 | 1 | 1.00 | 7.00 | ||||||
| 1 | 3 | 0.33 | -3.89 | ||||||
| 2 | 1 | 2.00 | 0.00 | k - 2 | |||||
| 2 | 3 | 0.67 | 3.56 | ||||||
| 4 | 1 | 4.00 | -32.00 | ||||||
| 4 | 3 | 1.33 | 7.11 | ||||||
| 8 | 1 | 8.00 | 336.00 | ||||||
| 8 | 3 | 2.67 | -12.44 | ||||||
| 16 | 1 | 16.00 | 6832.00 | ||||||
| 16 | 3 | 5.33 | -8.89 |
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
3k3 - 24k2 + 44k - 16
can be divided with k - 2
Polynomial Long Division :
12.5 Polynomial Long Division
Dividing : 3k3 - 24k2 + 44k - 16
("Dividend")
By : k - 2 ("Divisor")
| dividend | 3k3 | - | 24k2 | + | 44k | - | 16 | ||
| - divisor | * 3k2 | 3k3 | - | 6k2 | |||||
| remainder | - | 18k2 | + | 44k | - | 16 | |||
| - divisor | * -18k1 | - | 18k2 | + | 36k | ||||
| remainder | 8k | - | 16 | ||||||
| - divisor | * 8k0 | 8k | - | 16 | |||||
| remainder | 0 |
Quotient : 3k2-18k+8 Remainder: 0
Trying to factor by splitting the middle term
12.6 Factoring 3k2-18k+8
The first term is, 3k2 its coefficient is 3 .
The middle term is, -18k its coefficient is -18 .
The last term, "the constant", is +8
Step-1 : Multiply the coefficient of the first term by the constant 3 • 8 = 24
Step-2 : Find two factors of 24 whose sum equals the coefficient of the middle term, which is -18 .
| -24 | + | -1 | = | -25 | ||
| -12 | + | -2 | = | -14 | ||
| -8 | + | -3 | = | -11 | ||
| -6 | + | -4 | = | -10 | ||
| -4 | + | -6 | = | -10 | ||
| -3 | + | -8 | = | -11 | ||
| -2 | + | -12 | = | -14 | ||
| -1 | + | -24 | = | -25 | ||
| 1 | + | 24 | = | 25 | ||
| 2 | + | 12 | = | 14 | ||
| 3 | + | 8 | = | 11 | ||
| 4 | + | 6 | = | 10 | ||
| 6 | + | 4 | = | 10 | ||
| 8 | + | 3 | = | 11 | ||
| 12 | + | 2 | = | 14 | ||
| 24 | + | 1 | = | 25 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
-k • (3k2 - 18k + 8) • (k - 2)
——————————————————————————————
4
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