(3x3 + 3x2 + 5x - 1) • (x - 1)
——————————————————————————————
x + 1
Step-by-step explanation:Step by Step Solution:
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STEP
1
:
Equation at the end of step 1
STEP
2
:
Equation at the end of step
2
:
STEP
3
:
3x4 + 2x2 - 6x + 1
Simplify ——————————————————
x + 1
Checking for a perfect cube :
3.1 3x4 + 2x2 - 6x + 1 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 3x4 + 2x2 - 6x + 1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -6x + 1
Group 2: 3x4 + 2x2
Pull out from each group separately :
Group 1: (-6x + 1) • (1) = (6x - 1) • (-1)
Group 2: (3x2 + 2) • (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 :
3.3 Find roots (zeroes) of : F(x) = 3x4 + 2x2 - 6x + 1
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 1.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 12.00
-1 3 -0.33 3.26
1 1 1.00 0.00 x - 1
1 3 0.33 -0.74
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
3x4 + 2x2 - 6x + 1
can be divided with x - 1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 3x4 + 2x2 - 6x + 1
("Dividend")
By : x - 1 ("Divisor")
dividend 3x4 + 2x2 - 6x + 1
- divisor * 3x3 3x4 - 3x3
remainder 3x3 + 2x2 - 6x + 1
- divisor * 3x2 3x3 - 3x2
remainder 5x2 - 6x + 1
- divisor * 5x1 5x2 - 5x
remainder - x + 1
- divisor * -x0 - x + 1
remainder 0
Quotient : 3x3+3x2+5x-1 Remainder: 0
Polynomial Roots Calculator :
3.5 Find roots (zeroes) of : F(x) = 3x3+3x2+5x-1
See theory in step 3.3
In this case, the Leading Coefficient is 3 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -6.00
-1 3 -0.33 -2.44
1 1 1.00 10.00
1 3 0.33 1.11
Polynomial Roots Calculator found no rational roots
Polynomial Long Division :
3.6 Polynomial Long Division
Dividing : 3x3+3x2+5x-1
("Dividend")
By : x+1 ("Divisor")
dividend 3x3 + 3x2 + 5x - 1
- divisor * 3x2 3x3 + 3x2
remainder 5x - 1
- divisor * 0x1
remainder 5x - 1
- divisor * 5x0 5x + 5
remainder - 6
Quotient : 3x2+5
Remainder : -6
