Hurry please I will mark brainliest if you get it correct!! Evaluate the polynomial for the given value by using synthetic division. P(x) = 2x^3 + 7x^2 − 12 for x = −3 and x = 3

The Remainder Theorem states that when we divide a polynomial

f

(

x

)

by

x

−

c

the remainder

R

equals

f

(

c

)

.

We use synthetic division to divide

f

(

x

)

by

x

−

c

, where

c

=

−

3

.

Step 1. Write only the coefficients of

x

in the dividend inside an upside-down division symbol.

∣

1

−

4

2

−

4

+

6

∣

1

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

Step 2. Put the divisor at the left.

−

3

∣

1

−

4

2

−

4

+

6

∣

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

Step 3. Drop the first coefficient of the dividend below the division symbol.

−

3

∣

1

−

4

2

−

4

+

6

∣

—

—

—

—

—

—

—

—

—

—

—

—

—

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—

—

—

—

—

1

Step 4. Multiply the drop-down by the divisor, and put the result in the next column.

−

3

∣

1

−

4

2

−

4

6

|

−

3

————————————————————

1

Step 5. Add down the column.

−

3

∣

1

−

4

2

−

4

6

|

−

3

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

1

−

7

Step 6. Repeat Steps 4 and 5 until you can go no farther

−

3

∣

1

−

4

2

−

4

6

|

−

3

+

21

−

69

219

—

—

—

—

—

—

—

—

—

—

—

—

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—

—

—

—

—

—

—

1

−

7

23

−

73

225

The remainder is

225

, so

f

(

−

3

)

=

225

.

Check:

x

4

−

4

x

3

+

2

x

2

–

4

x

+

6

=

(

−

3

)

4

−

4

(

−

3

)

3

+

2

(

−

3

)

2

−

4

(

−

3

)

+

6

=

81

+

108

+

18

+

12

+

6

=

225

P(x) = 2x^3 + 7x^2 − 12 for x = −3 and x = 3

Step-by-step explanation: