Proposition 12.5.14. Polynomials in R[x] have both right distributivity across addition: (g(x) +r(x))p(x) = g(x)p(x) +r(x)p(x), and left distributivity across addition: p(x) (q(x) +r(x)) = p(x)q(x) + p(x)r(x).
Proposition 12.5.14. Polynomials in R[x] have both right distributivity across addition: (g(x) +r(x))p(x) = g(x)p(x) +r(x)p(x), and left distributivity across addition: p(x) (q(x) +r(x)) = p(x)q(x) + p(x)r(x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do Exercise 12.5.15 and please show step by step and explain
Transcribed Image Text:PROOF. Το show right distributivity, we have:
71
(q(z) + r(x))p(x) =(Σ+
j=0
-
-
||
-
max(n.)
Σ
j=0
max(n.) m
Σ Σ(bya + cal)*
j=0 =0
max(n,e) m
- Σ Σ(byzana + cjzatz)
j=0 t=0
mix(n,e) m
max(n.) m
Σ Σααρα + Σ Σφα
j=0
i=0
j=0 i=0
j=0
max(n.)
Σ (0;te)a) Samart
Σα
j=0
i=0
Gaj
max(nl)
TTL
Σ (bind tead)) Σαγα
j=0
i=0
ΤΕ 772
Σαμαρά
i=0
j=0 i=0
772
(bind test) Saint)
i=0
771
-ΣΣ+ΣΣτα
j=0 t=0
ι
ΤΟ
Τ
ΤΙΣ
-Στη Σα+Στη Σαμ
j=0
j=0
i=0
i=0
=q(x)p(a) + r(x)p(a),
which gives us right distributivity. We'll leave left distributivity up to you:
Exercise 12.5.15. Provide justification for each of the steps in the calcu-
lation in Proposition 12.5.14
![Proposition 12.5.14. Polynomials in R[x] have both right distributivity
across addition:
(q(x) +r(x))p(x) = q(x)p(x) +r(x)p(x),
and left distributivity across addition:
p(x) (q(x) +r(x)) = p(x)q(x) + p(x)r(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F39dc5b0c-4284-4639-9138-cb4cf09ffcc9%2Fvosnw8c_processed.png&w=3840&q=75)
Transcribed Image Text:Proposition 12.5.14. Polynomials in R[x] have both right distributivity
across addition:
(q(x) +r(x))p(x) = q(x)p(x) +r(x)p(x),
and left distributivity across addition:
p(x) (q(x) +r(x)) = p(x)q(x) + p(x)r(x).
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