Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(((d4)-(6•(d3)))+(22•3d2))-8dStep 2 :
Equation at the end of step 2 :
(((d4) - (2•3d3)) + (22•3d2)) - 8d
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
d4 - 6d3 + 12d2 - 8d =
d • (d3 - 6d2 + 12d - 8)
Checking for a perfect cube :
4.2 d3 - 6d2 + 12d - 8 is not a perfect cube
Trying to factor by pulling out :
4.3 Factoring: d3 - 6d2 + 12d - 8
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: d3 - 8
Group 2: -6d2 + 12d
Pull out from each group separately :
Group 1: (d3 - 8) • (1)
Group 2: (d - 2) • (-6d)
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 :
4.4 Find roots (zeroes) of : F(d) = d3 - 6d2 + 12d - 8
Polynomial Roots Calculator is a set of methods aimed at finding values of d for which F(d)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers d 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 | d - 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
d3 - 6d2 + 12d - 8
can be divided with d - 2
Polynomial Long Division :
4.5 Polynomial Long Division
Dividing : d3 - 6d2 + 12d - 8
("Dividend")
By : d - 2 ("Divisor")
| dividend | d3 | - | 6d2 | + | 12d | - | 8 | ||
| - divisor | * d2 | d3 | - | 2d2 | |||||
| remainder | - | 4d2 | + | 12d | - | 8 | |||
| - divisor | * -4d1 | - | 4d2 | + | 8d | ||||
| remainder | 4d | - | 8 | ||||||
| - divisor | * 4d0 | 4d | - | 8 | |||||
| remainder | 0 |
Quotient : d2-4d+4 Remainder: 0
Trying to factor by splitting the middle term
4.6 Factoring d2-4d+4
The first term is, d2 its coefficient is 1 .
The middle term is, -4d 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
d2 - 2d - 2d - 4
Step-4 : Add up the first 2 terms, pulling out like factors :
d • (d-2)
Add up the last 2 terms, pulling out common factors :
2 • (d-2)
Step-5 : Add up the four terms of step 4 :
(d-2) • (d-2)
Which is the desired factorization
Multiplying Exponential Expressions :
4.7 Multiply (d-2) by (d-2)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (d-2) and the exponents are :
1 , as (d-2) is the same number as (d-2)1
and 1 , as (d-2) is the same number as (d-2)1
The product is therefore, (d-2)(1+1) = (d-2)2
Multiplying Exponential Expressions :
4.8 Multiply (d-2)2 by (d-2)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (d-2) and the exponents are :
2
and 1 , as (d-2) is the same number as (d-2)1
The product is therefore, (d-2)(2+1) = (d-2)3
Final result :
d • (d - 2)3
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