# Remainder and Factor theorems

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### M4L4 The factor theorem and Remainder theorem:

M4L4 The factor theorem and Remainder theorem Adapted from Reynaldo B Pantino’s presentation

### Objectives::

Objectives: To identify whether a given linear binomial is a factor of a polynomial function. To determine the remainder if you divide by a linear binomial.

### Slide3:

Is 3 is a factor of 27? If yes, what makes it a factor of 27? It divides evenly with no remainder.

### Slide4:

The Remainder Theorem states that when the polynomial P(x) is divided by x – c, the remainder is P(c). Example: When P(x) = x 3 – x 2 – 4x + 4 is divided by x – 2, the remainder is 0. That is, P(2) = 0.

### Slide5:

The Remainder Theorem states that when the polynomial P(x) is divided by x – c, the remainder is P(c). Example: When P(x) = x 3 – x 2 – 4x + 4 is divided by x – 2, we have; 2 1 -1 -4 4 1 1 -2 0 2 2 -4 remainder

### Slide6:

Notice that P(c) = 0, using synthetic division P(x) = (x – c) ● Q(x) + R becomes P(x) = (x – c) ● Q(x) + 0 P(x) = (x – c) ● Q(x). 2 1 -1 -4 4 1 1 -2 0 2 2 -4 remainder

### Slide7:

FACTOR THEOREM Let P(x) be a polynomial. If P(c) = 0, where c is a real number, then (x – c) is a factor of P(x). Conversely, if (x – c) is a factor of P(x), then P(c) = 0. Since the theorem has a converse, the proof consists of two parts. a.) If (x – c) is a factor of P(x), then P(c) = 0. b.) If P(c) = 0, then (x – c) is a factor of P(x).

### Examples::

Examples: 1. Show that x + 1 is a factor of 2x 3 + 5x 2 – 3. Solution: Let P(x) = 2x 3 + 5x 2 – 3 P(-1) = 2(-1) 3 + 5(-1) 2 – 3 P(-1) = -2 + 5 – 3 P(-1) = 0 By the Remainder Theorem, we know there is no remainder because P(-1) = 0. By Factor theorem, x + 1 is a factor of 2x 3 + 5x 2 – 3 .

### Examples::

2. Check is x + 2 is a factor of x 4 + x 3 – x 2 – x - 18. Solution: Let P(x) = x 4 + x 3 – x 2 – x - 18 P(-2) = (-2) 4 + (-2) 3 – (-2) 2 – (-2) – 18 P(-2) = 16 - 8 – 4 + 2 – 18 P(-2) = -12 By the Remainder Theorem, we know the remainder is -12 because P(-2)=-12. By Factor theorem, x – 2 is a factor of x 4 + x 3 – x 2 – x - 18 . Examples:

### Examples::

3. Show that f(x) = x 3 + x 2 – 5x + 3 is divisible by x + 3. Solution : Let, f(-3) = (-3) 3 + (-3) 2 – 5 (-3) + 3 = 0 = -27 + 9 + 15 + 3 = -27 +27 = 0 By the Remainder Theorem, we know there is no remainder because f(-3) = 0. By Factor theorem, x + 3 is a factor of x 3 + x 2 – 5x + 3 meaning it is divisible by x+3 . Examples:

### PRACTICE EXERCISES:

PRACTICE EXERCISES