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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 12.5.15 and please show step by step and explain

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
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).
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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