factorise: viii) x3 + 3x2 - 7x - 6
please answer my question
Answers
STEP
1
:
Equation at the end of step 1
(((x3) + 3x2) - 7x) - 6
STEP
2
:
Checking for a perfect cube
2.1 x3+3x2-7x-6 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3+3x2-7x-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -7x-6
Group 2: x3+3x2
Pull out from each group separately :
Group 1: (7x+6) • (-1)
Group 2: (x+3) • (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+3x2-7x-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 ....
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+3x2-7x-6
can be divided with x-2
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3+3x2-7x-6
Quotient : x2+5x+3 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2+5x+3
The first term is, x2 its coefficient is 1 .
The middle term is, +5x its coefficient is 5 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant 1 • 3 = 3
Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is 5 .
-3 + -1 = -4
-1 + -3 = -4
1 + 3 = 4
3 + 1 = 4
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored