36x^3-5x^2-46x-5 please find this
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Answer:
Step by Step Solution:
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Step by step solution :
STEP
1
:
Equation at the end of step 1
(((x3) - 5x2) - 46x) - 40 = 0
STEP
2
:
Checking for a perfect cube
2.1 x3-5x2-46x-40 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-5x2-46x-40
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -46x-40
Group 2: x3-5x2
Pull out from each group separately :
Group 1: (23x+20) • (-2)
Group 2: (x-5) • (x2)
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 :
2.3 Find roots (zeroes) of : F(x) = x3-5x2-46x-40
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 1 and the Trailing Constant is -40.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,5 ,8 ,10 ,20 ,40
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x+1
-2 1 -2.00 24.00
-4 1 -4.00 0.00 x+4
-5 1 -5.00 -60.00
-8 1 -8.00 -504.00
-10 1 -10.00 -1080.00
-20 1 -20.00 -9120.00
-40 1 -40.00 -70200.00
1 1 1.00 -90.00
2 1 2.00 -144.00
4 1 4.00 -240.00
5 1 5.00 -270.00
8 1 8.00 -216.00
10 1 10.00 0.00 x-10
20 1 20.00 5040.00
40 1 40.00 54120.00
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
x3-5x2-46x-40
can be divided by 3 different polynomials,including by x-10
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-5x2-46x-40
("Dividend")
By : x-10 ("Divisor")
dividend x3 - 5x2 - 46x - 40
- divisor * x2 x3 - 10x2
remainder 5x2 - 46x - 40
- divisor * 5x1 5x2 - 50x
remainder 4x - 40
- divisor * 4x0 4x - 40
remainder 0
Quotient : x2+5x+4 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2+5x+4
The first term is, x2 its coefficient is 1 .
The middle term is, +5x its coefficient is 5 .
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 5 .
-4 + -1 = -5
-2 + -2 = -4
-1 + -4 = -5
1 + 4 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 4
x2 + 1x + 4x + 4
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+1)
Add up the last 2 terms, pulling out common factors :
4 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+4) • (x+1)
Which is the desired factorization
Equation at the end of step
2
:
(x + 4) • (x + 1) • (x - 10) = 0
STEP
3
:
Theory - Roots of a product
3.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Step-by-step explanation:
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