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UNIVERSITY OF DODOMA
COLLEGE OF EDUCATION
 DIPLOMA IN EDUCATION (SCIENCE, MATHS & ICT)
 MT 0124: ALGEBRA
ACADEMIC YEAR: 2014/2015

5.0 REMAINDER AND FACTOR THEOREMS
Aim: To determine the remainder when a polynomial Px is divided by a polynomial  x-c or by its multiple. i.e.  x-cQ(x)  where Q(x) is a polynomial.
Recall:
If a polynomial Px leave the remainder a polynomial rx when divided by a polynomial Sx, then the polynomials are in relation  Px=QxPx+rx Where  Px is called a dividend, Qx a quotient, Sx a divisor and rx a remainder.
In particular, when 10 is divided by 4 then 10=2×4+2.
When x3 is divided by x2-1 then
x3=x(x2-1)+x.
Synthetic division: is a method used when dividing a polynomial by (x – c) where (c) is a real number (positive or negative).
Remainder Theorem: If a polynomial P(x) is divided by x - c, then the remainder is P(c).
This is a case when Qx=x-c  in the said relation.
 Thus,
 Px=x-cSx+rx Which implies Pc=0×Sc+rc=r(c) which is the remainder when  x=c as asserted by the theorem.

This theorem gives us another way to evaluate a polynomial at x=c as we shall see.
Factor Theorem: If  Pc=0 then x - c is a factor of  P(x).
In the relation Px=x-cSx+rx;  Pc=0 implies rc=0 and the relation simplifies to Px=x-cSx which means x-c  is a factor of Px.
SYNTHETIC DIVISION
Steps for synthetic division when dividing P(x) by x - c:
 Synthetic division will consist of three rows
Write c and the coefficients of the dividend in descending order in the first row. If any x terms are missing, place a zero in its place.
Bring the leading coefficient in the top row down to the bottom (third) row.
Next, multiply the first number in the bottom row by c and place this product in the second row under the next coefficient and add these two terms together.
Continue this process until you reach the last column.
The numbers in the bottom row are the coefficients of the quotient and the remainder. The quotient will have one degree less than the dividend.
Example: Divide P(x) = 2x4 - 8x3 + 5x2 + 4 by x - 3.
First, note that the x term is missing so we must record a 0 in its place.
                                             
Therefore, the quotient is 2x3 -2x2 – x – 3 and the remainder is -5.
Common Mistakes to Avoid
DO NOT forget to record a zero for any missing terms. For example, suppose the dividend is f(x) = 3x4 - 5x2 - 2. Since both the x3 and x terms are missing we would record the coefficients as
3    0    -5   0    -2
Remember to add the terms inside the synthetic division. If the divisor is x + c, then the number outside the synthetic division is -c. For example, if the divisor is x + 5 then we record a -5 on the outside of the synthetic division.


PROBLEMS

LONG DIVISION OF POLYNOMIALS
Consider the polynomial below.
 Divide f(x) by x-2 and use the results to factor out the polynomial completely.

From this division, we have that 6x3 – 19x2+ 16x – 4 = (x – 2) (6x2 -7x +2) and by factoring 6x2 – 7x + 2  we have  6x3 – 19x2+ 16x – 4 = (x – 2) (2x – 1) (3x – 2).
Note that this factorization agrees with the graph shown in the figure above. That is, the graph intercepts the x- axis at three distinct points namely at x = 2, x =12, and x =23 .

Again; consider the operation below


This can be read as  

This implies,

Example: Use the Remainder Theorem to evaluate the following function at x = –2.

P (x) = 3x3 + 8x2 + 5x – 7

Solution; using synthetic division:
 We can obtain the following;

Because the remainder is r = –9, you can conclude that f (-2) = -9………. r = P (c)
This means that (–2, –9) is a point on the graph of P. You can check this by substituting x = -2 in the original function.
 P (-2) = 3 (-2)3 + 8 (-2)2 + 5 (-2) – 7
P (-2) = -24 +32 -10 -7
           = -9
Example: Show that (x – 2) and (x + 3) are factors of f (x) = 2x4+ 7x3 – 4x2 – 27x – 18. Then find the remaining factors of f (x).
Solution: using synthetic division’
Starting with the factor (x -2);

Take the result of this division and perform synthetic division again using the factor (x + 3).



Because the resulting quadratic expression factors as 2x2 + 5x + 3 = (2x + 3)(x + 1) the complete factorization of f (x) is f (x) = (x – 2)(x + 3)(2x + 3)(x + 1).

Finally, let’s check the uses of a remainder in synthetic division


****QUESTIONS??****