Unknown

58 2. Evaluation, Division, and Expansion

If we think of the “size” of a polynomial as being measured by its de-

gree, then, as with numbers, we can take away a multiple of g(t) which

leaves as a remainder, a “smaller” quantity which will in turn be sub-

sequently divided by g(t). In the context of polynomials, “smaller”

means “of lower degree”. Subtracting 4t2g(t) will leave such a remain-

der.

(4

(b)

cc>

Verify that f(t) - 4t2g(t) = -31t4 - 12t3 + t2 + 6.

Take away a multiple of g(t) from the right hand side of (a) to

leave a remainder of lower degree. Continue on in this fashion

until a remainder of degree less than that of g(t) is obtained.

Show that the process of parts (a) and (b) can be written in the

form of the long division algorithm:

4tz - 31t + 205

t3 + 7tz + 3t - 21 4t5 - 3t4 + OP - 7P + Ot+ 6

’ 4t5 + 28t4 -i- 12t3 - 8t2

• 31t4 - 12t3 + t2 + Ot+ 6


• 31t4 - 217t3 - 93t2 + 62t
  205t3 -I- 94t2 - 62t + 6
  205t3 + 1435t2 + 615t - 410
  
  
  • 1341t2 - 677t + 416
  
  
  
  

(4

(e)

From the algorithm in (c), read off the quotient and the remain-

der for the equation

f(t) = g(W) + r(t).

Check your answer by directly computing the right hand side.

The algorithm in (c) can be more clearly presented by suppress-

ing the variable to obtain

4 -31 205

173 -2)4 -3 0 -7 0 6

4 28 12 -8

-31 -12 1

-31 -217 -93 62

205 94 -62

205 1435 615 -410

-1341 -677 416

A further compression of this is Homer’s Method of Synthetic Di-

vision which can be regarded as a generalization of his method for

division by a binomial (t - c) and which in this case takes the form: