5a3 + 11a2 + 4a - 2
Answers
Step-by-step explanation:
Changes made to your input should not affect the solution:
(1): "a2" was replaced by "a^2". 1 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
(((5 • (a3)) + 11a2) + 4a) - 2
STEP
2
:
Equation at the end of step
2
:
((5a3 + 11a2) + 4a) - 2
STEP
3
:
Checking for a perfect cube
3.1 5a3+11a2+4a-2 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 5a3+11a2+4a-2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 4a-2
Group 2: 5a3+11a2
Pull out from each group separately :
Group 1: (2a-1) • (2)
Group 2: (5a+11) • (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) = 5a3+11a2+4a-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 5 and the Trailing Constant is -2.
The factor(s) are:
of the Leading Coefficient : 1,5
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 a+1
-1 5 -0.20 -2.40
-2 1 -2.00 -6.00
-2 5 -0.40 -2.16
1 1 1.00 18.00
1 5 0.20 -0.72
2 1 2.00 90.00
2 5 0.40 1.68
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
5a3+11a2+4a-2
can be divided with a+1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 5a3+11a2+4a-2
("Dividend")
By : a+1 ("Divisor")
dividend 5a3 + 11a2 + 4a - 2
- divisor * 5a2 5a3 + 5a2
remainder 6a2 + 4a - 2
- divisor * 6a1 6a2 + 6a
remainder - 2a - 2
- divisor * -2a0 - 2a - 2
remainder 0
Quotient : 5a2+6a-2 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 5a2+6a-2
The first term is, 5a2 its coefficient is 5 .
The middle term is, +6a its coefficient is 6 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 5 • -2 = -10
Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is 6 .
-10 + 1 = -9
-5 + 2 = -3
-2 + 5 = 3
-1 + 10 = 9
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(5a2 + 6a - 2) • (a + 1)
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