factorise : 4a3-9a² +3a+2
Answers
Answer:
STEP
1
:
Equation at the end of step 1
(((4 • (a3)) - 32a2) + 3a) + 2
STEP
2
:
Equation at the end of step
2
:
((22a3 - 32a2) + 3a) + 2
STEP
3
:
Checking for a perfect cube
3.1 4a3-9a2+3a+2 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 4a3-9a2+3a+2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3a+2
Group 2: -9a2+4a3
Pull out from each group separately :
Group 1: (3a+2) • (1)
Group 2: (4a-9) • (a2)
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 :
3.3 Find roots (zeroes) of : F(a) = 4a3-9a2+3a+2
Polynomial Roots Calculator is a set of methods aimed at finding values of a for which F(a)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers a 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 4 and the Trailing Constant is 2.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -14.00
-1 2 -0.50 -2.25
-1 4 -0.25 0.62
-2 1 -2.00 -72.00
1 1 1.00 0.00 a-1
1 2 0.50 1.75
1 4 0.25 2.25
2 1 2.00 4.00
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
4a3-9a2+3a+2
can be divided with a-1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 4a3-9a2+3a+2
("Dividend")
By : a-1 ("Divisor")
dividend 4a3 - 9a2 + 3a + 2
- divisor * 4a2 4a3 - 4a2
remainder - 5a2 + 3a + 2
- divisor * -5a1 - 5a2 + 5a
remainder - 2a + 2
- divisor * -2a0 - 2a + 2
remainder 0
Quotient : 4a2-5a-2 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 4a2-5a-2
The first term is, 4a2 its coefficient is 4 .
The middle term is, -5a its coefficient is -5 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 4 • -2 = -8
Step-2 : Find two factors of -8 whose sum equals the coefficient of the middle term, which is -5 .
-8 + 1 = -7
-4 + 2 = -2
-2 + 4 = 2
-1 + 8 = 7
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(4a2 - 5a - 2) • (a - 1)