Algebra-2

The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), the remainder of this division is equal to f(c). This theorem is particularly useful for evaluating polynomials at specific points without performing the actual polynomial long division.

Let's go through the examples you provided to illustrate the application of the Remainder Theorem.

A.

Value of ( c ): c = 2

Using the Remainder Theorem:

We need to find f(2):

Calculating each term:

4(2) = 8

-6 remains as is.

Putting it all together:

f(2) = 40 - 8 + 8 - 6 = 34

Remainder:

The remainder when dividing f(x) by ( x - 2 ) is ( 34 ), not ( 0 ).

B.

Value of c: c = 1

Using the Remainder Theorem:

We find f(1):

Calculating each term:

-2(1) = -2

7 remains as is.

Putting it all together:

f(1) = 1 + 3 - 2 + 7 = 9

Remainder:

The remainder is 9.

C.

Value of c: c = 3

Using the Remainder Theorem:

We find f(3):

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