SOLUTION: I neeed help asap please. Use Synthetic division to find the 3 zeros of {{{x^3+7x^2+2x-40=0}}}

Algebra.Com
Question 123010: I neeed help asap please.
Use Synthetic division to find the 3 zeros of
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Use Synthetic division to find the 3 zeros of
------------------------------------------
-5)....1....7....2....-40
........1....2....-8...|..0
-4)......1....-2...|..0
2)........1....|..0
==========================
The zeroes are x=-5, x=-4, x=2
Cheers,
Stan H.
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients


So let's list the factors of -40 (the last coefficient):



Now let's list the factors of 1 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct rational zeros of the function that could occur




To save time, I'm only going to use synthetic division on the possible zeros that are actually zeros of the function.
Otherwise, I would have to use synthetic division on every possible root (there are 16 possible roots, so that means there would be at most 16 synthetic division tables).
However, you might be required to follow this procedure, so this is why I'm showing you how to set up a problem like this



When you graph this polynomial, you will see that is a zero. So we'll use this for the synthetic division




Now set up the synthetic division table by placing the zero in the upper left corner and placing the coefficients of the polynomial to the right of the test zero.
2|172-40
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
2|172-40
|
1

Multiply 2 by 1 and place the product (which is 2) right underneath the second coefficient (which is 7)
2|172-40
|2
1

Add 2 and 7 to get 9. Place the sum right underneath 2.
2|172-40
|2
19

Multiply 2 by 9 and place the product (which is 18) right underneath the third coefficient (which is 2)
2|172-40
|218
19

Add 18 and 2 to get 20. Place the sum right underneath 18.
2|172-40
|218
1920

Multiply 2 by 20 and place the product (which is 40) right underneath the fourth coefficient (which is -40)
2|172-40
|21840
1920

Add 40 and -40 to get 0. Place the sum right underneath 40.
2|172-40
|21840
19200

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,9,20) form the quotient




So

You can use this online polynomial division calculator to check your work

Basically factors to

Now lets break down further





Looking at we can see that the first term is and the last term is where the coefficients are 1 and 20 respectively.

Now multiply the first coefficient 1 and the last coefficient 20 to get 20. Now what two numbers multiply to 20 and add to the middle coefficient 9? Let's list all of the factors of 20:



Factors of 20:
1,2,4,5,10,20

-1,-2,-4,-5,-10,-20 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 20
1*20
2*10
4*5
(-1)*(-20)
(-2)*(-10)
(-4)*(-5)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 9

First NumberSecond NumberSum
1201+20=21
2102+10=12
454+5=9
-1-20-1+(-20)=-21
-2-10-2+(-10)=-12
-4-5-4+(-5)=-9



From this list we can see that 4 and 5 add up to 9 and multiply to 20


Now looking at the expression , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms


So factors to


Now reintroduce the factor





Now set each factor equal to zero:

, or

Now solve for x for each factor:

, or





------------------------------------------------------------



Answer:



So the zeros of are , or

RELATED QUESTIONS

Use Synthetic Division to find the 3 zeros of... (answered by scott8148)
Question: 3 is a zero of f(x)=2x^3-3x^2-11x+6. Use synthetic division to find the zeros... (answered by ankor@dixie-net.com)
list all rational zeros use synthetic division to test the possible rational roots and... (answered by shorty4aboo)
Could someone please help? Use synthetic division and the Remainder Theorem to find... (answered by stanbon)
Use synthetic division to find the upper bound of polynomial (x^4-x^3-7x^2+x+6) please... (answered by lwsshak3)
Please help me solve this question: 1. Use the Rational Theorem and synthetic division (answered by lwsshak3)
4.4 Use synthetic division to divide f(x) =x^3 +7x^2 -9x-63 by x+3. Use the result to... (answered by MathLover1)
Use synthetic division to find:... (answered by Nate,venugopalramana)
1. Use synthetic division to divide p(x) = x^4 - 10x^2 - 2x + 4 by x + 3 2. Let f(x) = (answered by josgarithmetic)