4 1. Fundamentals
(4) (1+ t2)-’
( r 1
3+t-2t=
t+7.- Let p(t) = 3t - 4 and q(t) = 2t2 - 5t + 8. Verify that
(a) (p + q)(t) = 2t2 - 2t f 4
(b) (7~ - 6q)(t) = -12t2 + 51t -^76
(c) (pq)(t) = (qp)(t) = 6t3 - 23t2 + 44t - 32
(d) (p o q)(t) = 6t2 - 15t + 20
(e) (q 0 p)(t) = 18t2 - 63t + 60.- In multiplying two polynomials together, we can use the method of
detached coefficients. In finding the product of the polynomials t3 +
3t2 - 2t + 4 and 2t2 + t + 6, the paper-and-pencil computation looks
like this:
1 3 -2 4
2 1 6
6 18 -12 24
13-2 4
2 6 -4 82 7 5 24 -8 24.
Justify this algorithm and use it to read off the product of the two
polynomials.- (a) Multiply th e polynomials 4t3 + 2t2 + 7t + 1 and 2t2 + t + 6 by
using the method of detached coefficients.
(b) Evaluate each of the two polynomials and their product at t =
(c) Compare the paper-and-pencil long multiplication for the prod-
uct of the numbers 4271 and 216 with the table given in (a).
- Using a pocket calculator, multiply 11254361 by 57762343 by each of
the following methods:
(a) Multiply the polynomials 1125t + 4361 and 5723 + 2343, and
evaluate the product at t = 104.
(b) Multiply the polynomials llt2+254t+361 and 57t2+762t+343,
and evaluate the product at t = 103.- Find the product of 26543645132 and 27568374445.
- Let p and q be nonzero polynomials. Show that