2x-6×x²-7x+6 multiply
Answers
Answer:
Step by Step Solution
solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
((2x3 - x2) - 7x) + 6
STEP
2
:
Checking for a perfect cube
2.1 2x3-x2-7x+6 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: 2x3-x2-7x+6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -7x+6
Group 2: 2x3-x2
Pull out from each group separately :
Group 1: (-7x+6) • (1) = (7x-6) • (-1)
Group 2: (2x-1) • (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) = 2x3-x2-7x+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 2 and the Trailing Constant is 6.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 10.00
-1 2 -0.50 9.00
-2 1 -2.00 0.00 x+2
-3 1 -3.00 -36.00
-3 2 -1.50 7.50
-6 1 -6.00 -420.00
1 1 1.00 0.00 x-1
1 2 0.50 2.50
2 1 2.00 4.00
3 1 3.00 30.00
3 2 1.50 0.00 2x-3
6 1 6.00 360.00
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
2x3-x2-7x+6
can be divided by 3 different polynomials,including by 2x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : 2x3-x2-7x+6
("Dividend")
By : 2x-3 ("Divisor")
dividend 2x3 - x2 - 7x + 6
- divisor * x2 2x3 - 3x2
remainder 2x2 - 7x + 6
- divisor * x1 2x2 - 3x
remainder - 4x + 6
- divisor * -2x0 - 4x + 6
remainder 0
Quotient : x2+x-2 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2+x-2
The first term is, x2 its coefficient is 1 .
The middle term is, +x its coefficient is 1 .
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 1 .
-2 + 1 = -1
-1 + 2 = 1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 2
x2 - 1x + 2x - 2
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 :
2 • (x-1)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x-1)
Which is the desired factorization
Final result :
(x + 2) • (x - 1) • (2x - 3)