synthetic division....?

Question

Use synthetic division to find the function values.
P(x) = x^3 + 7x^2 - 12x - 3; find P(-3), P(-2), and P(1)

Using synthetic division, determine whether the numbers are roots of the polynomials.
-4, 2; P(x)=3x^3 +11x^2 -2x +8


Please explain. Thank you in advance!!

Answer 1

synthetic division

1->put a column as shown below
2->write the coefficients of x^3 , x^2 . ... constant term (put zero if any term is missing in between


for x^3 + 7x^2 - 12x - 3


| 1 7 -12 -3
|
|
-----------------------------

put a zero below 1st term

put the value of x outside the column ( -3 in ur case)

in synthetic division u always add

multiply the result after addition with adding & continue till end

-3 | 1 7 -12 -3
|
| 0
-----------------------------
1




-3 | 1 7 -12 -3
|
| 0 -3
-----------------------------
1 4



-3 | 1 7 -12 -3
|
| 0 -3 -12
-----------------------------
1 4 -24




-3 | 1 7 -12 -3
|
| 0 -3 -12 72
-----------------------------
1 4 -24 69



so 69 is your answer


if u get ZERO in last place then x = value u put out of column(-3 here) is a root of the equation

or (x - value ) is a factor of the expression

for ex. -4 is a root of 3x^3 + 11x^2 - 2x +8

-4 | 3 11 -2 8
|
| 0 -12 4 -8
------------------------------------
3 -1 2 0


(note: if u divide the above expression by (x+4) then u get
3x^2 - x +2 ,u can get these coefficients from the table- see the added values - last row )

you can verify your answers by directly substituting in the expression

Answer 2

Synthetic Division is used for the second part of the question only.

P(-3) = (-3)³ + 63 + 36 - 3 = - 27 + 96 = 69
P(-2) = (-2)³ + 28 + 24 - 3 = - 8 + 49 = 41
P(1) = 1³ + 7 - 12 - 3 = 8 - 15 = -7

Test to see if -4 is a root:-

-4----|3-----11---- -2----- 8
------| ------ -12----- 4-- -8
------|3---- -1------- 2 --- 0 = R

R= 0 so x = -4 is a root.


Test to see if 2 is a root:-

2---|3-----11----- -2------8
-----|-------6------34-----64
-----3-----17-----32-----72=R

R does not = 0 so x = 2 is not a root.

Difficult and time consuming to show synthetic division but above is OK I hope.