reduce to its lowest term 3x²-27/2x²-5x-3
Answers
Answer:
STEP
1
:
Equation at the end of step 1
(3x2 - 27)
((—————————— • x2) - 5x) - 3
2
STEP
2
:
3x2 - 27
Simplify ————————
2
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
3x2 - 27 = 3 • (x2 - 9)
Trying to factor as a Difference of Squares:
3.2 Factoring: x2 - 9
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 9 is the square of 3
Check : x2 is the square of x1
Factorization is : (x + 3) • (x - 3)
Equation at the end of step
3
:
3•(x+3)•(x-3)
((—————————————•x2)-5x)-3
2
STEP
4
:
Equation at the end of step 4
3x2 • (x + 3) • (x - 3)
(——————————————————————— - 5x) - 3
2
STEP
5
:
Rewriting the whole as an Equivalent Fraction
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 2 as the denominator :
5x 5x • 2
5x = —— = ——————
1 2
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 :
5.2 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:
3x2 • (x+3) • (x-3) - (5x • 2) 3x4 - 27x2 - 10x
—————————————————————————————— = ————————————————
2 2
Equation at the end of step
5
:
(3x4 - 27x2 - 10x)
—————————————————— - 3
2
STEP
6
:
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 2 as the denominator :
3 3 • 2
3 = — = —————
1 2
STEP
7
:
Pulling out like terms :
7.1 Pull out like factors :
3x4 - 27x2 - 10x = x • (3x3 - 27x - 10)
Polynomial Roots Calculator :
7.2 Find roots (zeroes) of : F(x) = 3x3 - 27x - 10
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 -10.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,5 ,10
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 14.00
-1 3 -0.33 -1.11
-2 1 -2.00 20.00
-2 3 -0.67 7.11
-5 1 -5.00 -250.00
-5 3 -1.67 21.11
-10 1 -10.00 -2740.00
-10 3 -3.33 -31.11
1 1 1.00 -34.00
1 3 0.33 -18.89
2 1 2.00 -40.00
2 3 0.67 -27.11
5 1 5.00 230.00
5 3 1.67 -41.11
10 1 10.00 2720.00
10 3 3.33 11.11
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
7.3 Adding up the two equivalent fractions
x • (3x3-27x-10) - (3 • 2) 3x4 - 27x2 - 10x - 6
—————————————————————————— = ————————————————————
2 2
Checking for a perfect cube :
7.4 3x4 - 27x2 - 10x - 6 is not a perfect cube
Trying to factor by pulling out :
7.5 Factoring: 3x4 - 27x2 - 10x - 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -10x - 6
Group 2: 3x4 - 27x2
Pull out from each group separately :
Group 1: (5x + 3) • (-2)
Group 2: (x2 - 9) • (3x2)
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 :
7.6 Find roots (zeroes) of : F(x) = 3x4 - 27x2 - 10x - 6
See theory in step 7.2
In this case, the Leading Coefficient is 3 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -20.00
-1 3 -0.33 -5.63
-2 1 -2.00 -46.00
-2 3 -0.67 -10.74
-3 1 -3.00 24.00
-6 1 -6.00 2970.00
1 1 1.00 -40.00
1 3 0.33 -12.30
2 1 2.00 -86.00
2 3 0.67 -24.07
3 1 3.00 -36.00
6 1 6.00 2850.00
Polynomial Roots Calculator found no rational roots
Final result :
3x4 - 27x2 - 10x - 6
————————————————————
2