Find Remainder
6x3 + 7x2 - 15x + 4 divided by (x - 1)
Answers
Answer:
STEP
1
:
Equation at the end of step 1
STEP
2
:
Equation at the end of step
2
:
STEP
3
:
6x3 + 7x2 - 15x + 6
Simplify ———————————————————
2x - 1
Checking for a perfect cube :
3.1 6x3 + 7x2 - 15x + 6 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 6x3 + 7x2 - 15x + 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -15x + 6
Group 2: 6x3 + 7x2
Pull out from each group separately :
Group 1: (5x - 2) • (-3)
Group 2: (6x + 7) • (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) = 6x3 + 7x2 - 15x + 6
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 6 and the Trailing Constant is 6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 22.00
-1 2 -0.50 14.50
-1 3 -0.33 11.56
-1 6 -0.17 8.67
-2 1 -2.00 16.00
Note - For tidiness, printing of 13 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 6x3 + 7x2 - 15x + 6
("Dividend")
By : 2x - 1 ("Divisor")
dividend 6x3 + 7x2 - 15x + 6
- divisor * 3x2 6x3 - 3x2
remainder 10x2 - 15x + 6
- divisor * 5x1 10x2 - 5x
remainder - 10x + 6
- divisor * -5x0 - 10x + 5
remainder 1
Quotient : 3x2 + 5x - 5
Remainder : 1
Final result :
6x3 + 7x2 - 15x + 6
———————————————————
2x - 1
See results of polynomial long division:
1. In step #03.04