2x^3-x^2-5x+2 factorise by division method
Answers
[tex]\bold\red{ANSWER}[\tex]
Factoring: 2x3-x2-5x-2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -5x-2
Group 2: 2x3-x2
Pull out from each group separately :
Group 1: (5x+2) • (-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.
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 -2.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x+1
-1 2 -0.50 0.00 2x+1
-2 1 -2.00 -12.00
1 1 1.00 -6.00
1 2 0.50 -4.50
2 1 2.00 0.00 x-2
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-x2-5x-2
can be divided by 3 different polynomials,including by x-2
Find roots (zeroes) of : F(x) = 2x3-x2-5x-2
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
Polynomial Long Division
Dividing : 2x3-x2-5x-2
("Dividend")
By : x-2 ("Divisor")
dividend 2x3 - x2 - 5x - 2
- divisor * 2x2 2x3 - 4x2
remainder 3x2 - 5x - 2
- divisor * 3x1 3x2 - 6x
remainder x - 2
- divisor * x0 x - 2
remainder 0
Quotient : 2x2+3x+1 Remainder: 0
Factoring 2x2+3x+1
The first term is, 2x2 its coefficient is 2 .
The middle term is, +3x its coefficient is 3 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 2 • 1 = 2
Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 3 .
-2 + -1 = -3
-1 + -2 = -3
1 + 2 = 3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 2
2x2 + 1x + 2x + 1
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (2x+1)
Add up the last 2 terms, pulling out common factors :
1 • (2x+1)
Step-5 : Add up the four terms of step 4 :
(x+1) • (2x+1)
Which is the desired factorization
Final result :
(2x + 1) • (x + 1) • (x - 2)