Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(((x4)+(8•(x3)))+5x2)-50xStep 2 :
Equation at the end of step 2 :
(((x4) + 23x3) + 5x2) - 50x
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
x4 + 8x3 + 5x2 - 50x =
x • (x3 + 8x2 + 5x - 50)
Checking for a perfect cube :
4.2 x3 + 8x2 + 5x - 50 is not a perfect cube
Trying to factor by pulling out :
4.3 Factoring: x3 + 8x2 + 5x - 50
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 5x - 50
Group 2: x3 + 8x2
Pull out from each group separately :
Group 1: (x - 10) • (5)
Group 2: (x + 8) • (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 :
4.4 Find roots (zeroes) of : F(x) = x3 + 8x2 + 5x - 50
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 -50.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,5 ,10 ,25 ,50
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -48.00 | ||||||
| -2 | 1 | -2.00 | -36.00 | ||||||
| -5 | 1 | -5.00 | 0.00 | x + 5 | |||||
| -10 | 1 | -10.00 | -300.00 | ||||||
| -25 | 1 | -25.00 | -10800.00 | ||||||
| -50 | 1 | -50.00 | -105300.00 | ||||||
| 1 | 1 | 1.00 | -36.00 | ||||||
| 2 | 1 | 2.00 | 0.00 | x - 2 | |||||
| 5 | 1 | 5.00 | 300.00 | ||||||
| 10 | 1 | 10.00 | 1800.00 | ||||||
| 25 | 1 | 25.00 | 20700.00 | ||||||
| 50 | 1 | 50.00 | 145200.00 |
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 + 8x2 + 5x - 50
can be divided by 2 different polynomials,including by x - 2
Polynomial Long Division :
4.5 Polynomial Long Division
Dividing : x3 + 8x2 + 5x - 50
("Dividend")
By : x - 2 ("Divisor")
| dividend | x3 | + | 8x2 | + | 5x | - | 50 | ||
| - divisor | * x2 | x3 | - | 2x2 | |||||
| remainder | 10x2 | + | 5x | - | 50 | ||||
| - divisor | * 10x1 | 10x2 | - | 20x | |||||
| remainder | 25x | - | 50 | ||||||
| - divisor | * 25x0 | 25x | - | 50 | |||||
| remainder | 0 |
Quotient : x2+10x+25 Remainder: 0
Trying to factor by splitting the middle term
4.6 Factoring x2+10x+25
The first term is, x2 its coefficient is 1 .
The middle term is, +10x its coefficient is 10 .
The last term, "the constant", is +25
Step-1 : Multiply the coefficient of the first term by the constant 1 • 25 = 25
Step-2 : Find two factors of 25 whose sum equals the coefficient of the middle term, which is 10 .
| -25 | + | -1 | = | -26 | ||
| -5 | + | -5 | = | -10 | ||
| -1 | + | -25 | = | -26 | ||
| 1 | + | 25 | = | 26 | ||
| 5 | + | 5 | = | 10 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 5 and 5
x2 + 5x + 5x + 25
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+5)
Add up the last 2 terms, pulling out common factors :
5 • (x+5)
Step-5 : Add up the four terms of step 4 :
(x+5) • (x+5)
Which is the desired factorization
Multiplying Exponential Expressions :
4.7 Multiply (x+5) by (x+5)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x+5) and the exponents are :
1 , as (x+5) is the same number as (x+5)1
and 1 , as (x+5) is the same number as (x+5)1
The product is therefore, (x+5)(1+1) = (x+5)2
Final result :
x • (x + 5)2 • (x - 2)
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