Respuesta :
Step-by-step explanation:
Step 1 :Equation at the end of step 1
(((x3) + 7x2) + 4x) - 12
Step 2 :Checking for a perfect cube
2.1 x3+7x2+4x-12 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3+7x2+4x-12
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3+7x2
Group 2: 4x-12
Pull out from each group separately :
Group 1: (x+7) • (x2)
Group 2: (x-3) • (4)
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+7x2+4x-12
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 -12.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -10.00
-2 1 -2.00 0.00 x+2
-3 1 -3.00 12.00
-4 1 -4.00 20.00
-6 1 -6.00 0.00 x+6
-12 1 -12.00 -780.00
1 1 1.00 0.00 x-1
2 1 2.00 32.00
3 1 3.00 90.00
4 1 4.00 180.00
6 1 6.00 480.00
12 1 12.00 2772.00
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+7x2+4x-12
can be divided by 3 different polynomials,including by x-1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3+7x2+4x-12
("Dividend")
By : x-1 ("Divisor")
dividend x3 + 7x2 + 4x - 12
- divisor * x2 x3 - x2
remainder 8x2 + 4x - 12
- divisor * 8x1 8x2 - 8x
remainder 12x - 12
- divisor * 12x0 12x - 12
remainder 0
Quotient : x2+8x+12 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2+8x+12
The first term is, x2 its coefficient is 1 .
The middle term is, +8x its coefficient is 8 .
The last term, "the constant", is +12
Step-1 : Multiply the coefficient of the first term by the constant 1 • 12 = 12
Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is 8 .
-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7
-3 + -4 = -7
-2 + -6 = -8
-1 + -12 = -13
1 + 12 = 13
2 + 6 = 8 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 6
x2 + 2x + 6x + 12
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+2)
Add up the last 2 terms, pulling out common factors :
6 • (x+2)
Step-5 : Add up the four terms of step 4 :
(x+6) • (x+2)
Which is the desired factorization
Final result :
(x + 6) • (x + 2) • (x - 1)
