Step-by-step explanation:
STEP
1
:
Equation at the end of step 1
STEP
2
:
Equation at the end of step
2
:
STEP
3
:
10x3 - 3x2 - 7x + 3
Simplify ———————————————————
2x - 1
Checking for a perfect cube :
3.1 10x3 - 3x2 - 7x + 3 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 10x3 - 3x2 - 7x + 3
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -7x + 3
Group 2: 10x3 - 3x2
Pull out from each group separately :
Group 1: (-7x + 3) • (1) = (7x - 3) • (-1)
Group 2: (10x - 3) • (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) = 10x3 - 3x2 - 7x + 3
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 10 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2 ,5 ,10
of the Trailing Constant : 1 ,3
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -3.00
-1 2 -0.50 4.50
-1 5 -0.20 4.20
-1 10 -0.10 3.66
-3 1 -3.00 -273.00
-3 2 -1.50 -27.00
-3 5 -0.60 3.96
-3 10 -0.30 4.56
1 1 1.00 3.00
1 2 0.50 0.00 2x - 1
1 5 0.20 1.56
1 10 0.10 2.28
3 1 3.00 225.00
3 2 1.50 19.50
3 5 0.60 -0.12
3 10 0.30 0.90
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
10x3 - 3x2 - 7x + 3
can be divided with 2x - 1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 10x3 - 3x2 - 7x + 3
("Dividend")
By : 2x - 1 ("Divisor")
dividend 10x3 - 3x2 - 7x + 3
- divisor * 5x2 10x3 - 5x2
remainder 2x2 - 7x + 3
- divisor * x1 2x2 - x
remainder - 6x + 3
- divisor * -3x0 - 6x + 3
remainder 0
Quotient : 5x2+x-3 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 5x2+x-3
The first term is, 5x2 its coefficient is 5 .
The middle term is, +x its coefficient is 1 .
The last term, "the constant", is -3
Step-1 : Multiply the coefficient of the first term by the constant 5 • -3 = -15
Step-2 : Find two factors of -15 whose sum equals the coefficient of the middle term, which is 1 .
-15 + 1 = -14
-5 + 3 = -2
-3 + 5 = 2
-1 + 15 = 14
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Canceling Out :
3.6 Cancel out (2x-1) which appears on both sides of the fraction line.
Final result :
5x2 + x - 3
