Factorise 3x^3 - 23x^2 + 44x - 20 using factor theorem
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Answers
Step 1 :
20
Simplify ——
x
Equation at the end of step 1 :
20
((((3•(x3))-(23•(x2)))+44x)-——)-5
x
Step 2 :
Equation at the end of step 2 :
20
((((3•(x3))-23x2)+44x)-——)-5
x
Step 3 :
Equation at the end of step 3 :
20
(((3x3 - 23x2) + 44x) - ——) - 5
x
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x as the denominator :
3x3 - 23x2 + 44x (3x3 - 23x2 + 44x) • x
3x3 - 23x2 + 44x = ———————————————— = ——————————————————————
1 x
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 5 :
Pulling out like terms :
5.1 Pull out like factors :
3x3 - 23x2 + 44x = x • (3x2 - 23x + 44)
Trying to factor by splitting the middle term
5.2 Factoring 3x2 - 23x + 44
The first term is, 3x2 its coefficient is 3 .
The middle term is, -23x its coefficient is -23 .
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 -23 .
-132 + -1 = -133
-66 + -2 = -68
-44 + -3 = -47
-33 + -4 = -37
-22 + -6 = -28
-12 + -11 = -23 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -12 and -11
3x2 - 12x - 11x - 44
Step-4 : Add up the first 2 terms, pulling out like factors :
3x • (x-4)
Add up the last 2 terms, pulling out common factors :
11 • (x-4)
Step-5 : Add up the four terms of step 4 :
(3x-11) • (x-4)
Which is the desired factorization
Adding fractions that have a common denominator :
5.3 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:
x • (x-4) • (3x-11) • x - (20) 3x4 - 23x3 + 44x2 - 20
—————————————————————————————— = ——————————————————————
x x
Equation at the end of step 5 :
(3x4 - 23x3 + 44x2 - 20)
———————————————————————— - 5
x
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x as the denominator :
5 5 • x
5 = — = —————
1 x
Checking for a perfect cube :
6.2 3x4 - 23x3 + 44x2 - 20 is not a perfect cube
Trying to factor by pulling out :
6.3 Factoring: 3x4 - 23x3 + 44x2 - 20
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 44x2 - 20
Group 2: 3x4 - 23x3
Pull out from each group separately :
Group 1: (11x2 - 5) • (4)
Group 2: (3x - 23) • (x3)
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 :
6.4 Find roots (zeroes) of : F(x) = 3x4 - 23x3 + 44x2 - 20
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 -20.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4 ,5 ,10 ,20
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 50.00
-1 3 -0.33 -14.22
-2 1 -2.00 388.00
-2 3 -0.67 6.96
-4 1 -4.00 2924.00
Note - For tidiness, printing of 19 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
(3x4-23x3+44x2-20) - (5 • x) 3x4 - 23x3 + 44x2 - 5x - 20
———————————————————————————— = ———————————————————————————
x x
Polynomial Roots Calculator :
6.6 Find roots (zeroes) of : F(x) = 3x4 - 23x3 + 44x2 - 5x - 20
See theory in step 6.4
In this case, the Leading Coefficient is 3 and the Trailing Constant is -20.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4 ,5 ,10 ,20
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 55.00
-1 3 -0.33 -12.56
-2 1 -2.00 398.00
-2 3 -0.67 10.30
-4 1 -4.00 2944.00
Note - For tidiness, printing of 19 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Final result :
3x4 - 23x3 + 44x2 - 5x - 20
———————————————————————————
x
Answer:
3x2-75
Step 1 : Equation at the end of step 1 : 3x2 - 75.
Step 2 :
Step 3 : Pulling out like terms : 3.1 Pull out like factors : 3x2 - 75 = 3 • (x2 - 25) Trying to factor as a Difference of Squares : 3.2 Factoring: x2 - 25. Check : 25 is the square of 5. Check : x2 is the square of x1 Factorization is : (x + 5) • (x - 5)
Step-by-step explanation:
inbthis way you can solve that question