Write the Quotient and Remainder when we divide (x3 –6x2 +11x–6)by(x2 -5x+6)
Answers
Fully solved solution here;)
(x3-6x2+11x-6):(x-3) Final result : (x - 1) • (x - 2) Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step by step solution :Step 1 :Equation at the end of step 1 : Step 2 : x3 - 6x2 + 11x - 6 Simplify —————————————————— x - 3 Checking for a perfect cube :
2.1 x3 - 6x2 + 11x - 6 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3 - 6x2 + 11x - 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 11x - 6
Group 2: -6x2 + x3
Pull out from each group separately :
Group 1: (11x - 6) • (1)
Group 2: (x - 6) • (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 - 6x2 + 11x - 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 1 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 -24.00 -2 1 -2.00 -60.00 -3 1 -3.00 -120.00 -6 1 -6.00 -504.00 1 1 1.00 0.00 x - 1 2 1 2.00 0.00 x - 2 3 1 3.00 0.00 x - 3 6 1 6.00 60.00
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 - 6x2 + 11x - 6
can be divided by 3 different polynomials,including by x - 3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3 - 6x2 + 11x - 6
("Dividend")
By : x - 3 ("Divisor")
dividend x3 - 6x2 + 11x - 6 - divisor * x2 x3 - 3x2 remainder - 3x2 + 11x - 6 - divisor * -3x1 - 3x2 + 9x remainder 2x - 6 - divisor * 2x0 2x - 6 remainder 0
Quotient : x2-3x+2 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-3x+2
The first term is, x2 its coefficient is 1 .
The middle term is, -3x its coefficient is -3 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 1 • 2 = 2
Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is -3 .
-2 + -1 = -3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and -1
x2 - 2x - 1x - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-2)
Add up the last 2 terms, pulling out common factors :
1 • (x-2)
Step-5 : Add up the four terms of step 4 :
(x-1) • (x-2)
Which is the desired factorization
Canceling Out :
2.6 Cancel out (x-3) which appears on both sides of the fraction line.
Final result : (x - 1) • (x - 2)