(ii) (a) x3 + 4x2 – 5x - 6
Answers
Answer:
Step by Step Solution
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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
1
:
Equation at the end of step 1
(((x3) - 22x2) + 5x) - 6
STEP
2
:
Checking for a perfect cube
2.1 x3-4x2+5x-6 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-4x2+5x-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 5x-6
Group 2: x3-4x2
Pull out from each group separately :
Group 1: (5x-6) • (1)
Group 2: (x-4) • (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) = x3-4x2+5x-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 -16.00
-2 1 -2.00 -40.00
-3 1 -3.00 -84.00
-6 1 -6.00 -396.00
1 1 1.00 -4.00
2 1 2.00 -4.00
3 1 3.00 0.00 x-3
6 1 6.00 96.00
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
x3-4x2+5x-6
can be divided with x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-4x2+5x-6
("Dividend")
By : x-3 ("Divisor")
dividend x3 - 4x2 + 5x - 6
- divisor * x2 x3 - 3x2
remainder - x2 + 5x - 6
- divisor * -x1 - x2 + 3x
remainder 2x - 6
- divisor * 2x0 2x - 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 = -3
-1 + -2 = -3
1 + 2 = 3
2 + 1 = 3
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(x2 - x + 2) • (x - 3)