Question

state remainder theorem and factor theorem​

Answer 1

Answer:

The Remainder Theorem

Consider f(x) = (x − r)q(x) + R

Note that if we let x = r, the expression becomes

f(r) = (r − r) q(r) + R

Simplifying gives:

f(r) = R

This leads us to the Remainder Theorem which states:

If a polynomial f(x) is divided by (x − r) and a remainder R is obtained, then f(r) = R.

Example 3

Use the remainder theorem to find the remainder for Example 1 above, which was divide f(x) = 3x2 + 5x − 8 by (x − 2).

Example 4

By using the remainder theorem, determine the remainder when

3x3 − x2 − 20x + 5

is divided by (x + 4).

The Factor Theorem

The Factor Theorem states:

If the remainder f(r) = R = 0, then (x − r) is a factor of f(x).

The Factor Theorem is powerful because it can be used to find roots of polynomial equations.

Example 5

Is (x + 1) a factor of f(x) = x3 + 2x2 − 5x − 6?

Exercises

1. Find the remainder R by long division and by the Remainder Theorem.

(2x4 − 10x2 + 30x - 60) ÷ (x + 4)

2. Find the remainder using the Remainder Theorem

(x4 − 5x3 + x2 − 2x + 6) ÷ (x + 4)

3. Use the Factor Theorem to decide if (x − 2) is a factor of

f(x) = x5 − 2x4 + 3x3 − 6x2 − 4x + 8.

4. Determine whether \displaystyle-\frac{3}{{2}}−

2

3

is a zero (root) of the function:

f(x) = 2x3 + 3x2 − 8x − 12

Answer 2

Remainder theorem- Let p(x) be any polynomial of degree greater than or equal to one and let 'a' be
any real number. If p(x) is divided by the linear polynomial x - a, then the remainder is p(a).
Factor theorem- lf p(x) is a polynomial of degree greater and equal to 1 and a is any real number then (1) x -a is a factor of p(x), if p(a)=0, and (2) p(a)=0
, if x-a is a factor of p(x).
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