Respuesta :
Answer: (x-2)^3
Step-by-step explanation:
Frist step; (((x3) - (2•3x^2)) + 12x) - 8
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second step; x^3-6x^2+12x-8 is not a perfect cube
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third step; Factoring: x^3-6x^2+12x-8
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x^3-8
Group 2: -6x^2+12x
Pull out from each group separately :
Group 1: (x^3-8) • (1)
Group 2: (x-2) • (-6x)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
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step four;
Find roots (zeroes) of : F(x) = x^3-6x^2+12x-8
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 -8.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -27.00
-2 1 -2.00 -64.00
-4 1 -4.00 -216.00
-8 1 -8.00 -1000.00
1 1 1.00 -1.00
2 1 2.00 0.00 x-2
4 1 4.00 8.00
8 1 8.00 216.00
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
x^3-6x^2+12x-8
can be divided with x-2
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Step 5; Polynomial Long Division
Dividing : x^3-6x^2+12x-8
("Dividend")
By : x-2 ("Divisor")
dividend x3 - 6x2 + 12x - 8
- divisor * x^2 x^3 - 2x^2
remainder - 4x^2 + 12x - 8
- divisor * -4x^1 - 4x^2 + 8x
remainder 4x - 8
- divisor * 4x^0 4x - 8
remainder 0
Quotient : x^2-4x+4 Remainder: 0
Graph the cubic using its end behavior and a few selected points.
Falls to the left and rises to the right
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stp6;
Factoring x2-4x+4
The first term is, x2 its coefficient is 1 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -4 .
-4 + -1 = -5
-2 + -2 = -4 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and -2
x2 - 2x - 2x - 4
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 :
2 • (x-2)
Step-5 : Add up the four terms of step 4 :
(x-2) • (x-2)
Which is the desired factorization
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2.6 Multiply (x-2) by (x-2)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-2) and the exponents are :
1 , as (x-2) is the same number as (x-2)1
and 1 , as (x-2) is the same number as (x-2)1
The product is therefore, (x-2)(1+1) = (x-2)2
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Multiply (x-2)2 by (x-2)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-2) and the exponents are :
2
and 1 , as (x-2) is the same number as (x-2)1
The product is therefore, (x-2)(2+1) = (x-2)3
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Answer; (x - 2)^3
