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Factorize the following polynomials using synthetic division method
b) 2x^3- x^2 - 15x + 18
Answers
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
0-(2*x^3-x^2-15*x+18)=0
Step by step solution :
STEP1:Equation at the end of step 1
0 - (((2x3 - x2) - 15x) + 18) = 0
STEP2:Checking for a perfect cube
2.1 -2x3+x2+15x-18 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: -2x3+x2+15x-18
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 15x-18
Group 2: -2x3+x2
Pull out from each group separately :
Group 1: (5x-6) • (3)
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+15x-18
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 -18.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,3 ,6 ,9 ,18
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 -30.00 -1 2 -0.50 -25.00 -2 1 -2.00 -28.00 -3 1 -3.00 0.00 x+3 -3 2 -1.50 -31.50 -6 1 -6.00 360.00 -9 1 -9.00 1386.00 -9 2 -4.50 117.00 -18 1 -18.00 11700.00 1 1 1.00 -4.00 1 2 0.50 -10.50 2 1 2.00 0.00 x-2 3 1 3.00 -18.00 3 2 1.50 0.00 2x-3 6 1 6.00 -324.00 9 1 9.00 -1260.00 9 2 4.50 -112.50 18 1 18.00 -11088.00
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+15x-18
can be divided by 3 different polynomials,including by 2x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : -2x3+x2+15x-18
("Dividend")
By : 2x-3 ("Divisor")
dividend- 2x3 + x2 + 15x - 18 - divisor * -x2 - 2x3 + 3x2 remainder - 2x2 + 15x - 18 - divisor * -x1 - 2x2 + 3x remainder 12x - 18 - divisor * 6x0 12x - 18 remainder 0
Quotient : -x2-x+6 Remainder: 0
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