10.2. Let f(x) = 3x³ + x* + x³ + 3x? + 2x +4 and g(x) = 2x³ + 3x² + x +1 be polynomials in Zs[x]. Find q(x), r(x) e Zs[x], with deg(r(x)) < 3, such that f (x) = g(x)q(x)+r(x).
10.2. Let f(x) = 3x³ + x* + x³ + 3x? + 2x +4 and g(x) = 2x³ + 3x² + x +1 be polynomials in Zs[x]. Find q(x), r(x) e Zs[x], with deg(r(x)) < 3, such that f (x) = g(x)q(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
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Could you explain how to show 10.2 in great detail? I also included a list of theorems and definitions in my textbook as a reference. Really appreciate your help!
![Definition 10.1. Let R be a ring. Then a polynomial with coefficients in R is a
formal expression
ao + ajx + a2x² + · · · + anx",
where a; e Rand n is a nonnegative integer. Suppose that bo +bjx + • ·.+ bmx™
is also a polynomial with coefficients in R. Without loss of generality, let us say that
n < m. Then these polynomials are equal if and only if a; = b; for all i <n and
b; = 0 for alli > n. The set of all polynomials with coefficients in R is denoted
R[x].
Example 10.1. Let R = Z5. Then (inserting congruence class brackets for clarity),
an example of a polynomial in R[x] would be f (x) = [3] + [2]x + [4]x². As part
of the above definition, we observe that f (x) = g(x), where g(x) = [3] + [2]x +
[4]x? + [0]x³.
= ao + a¡x + · · · + a,,x" e R[x].
Definition 10.2. Let R be a ring and let f(x)
Further suppose that am +0 but ag = 0 for all k > m. Then the degree of f (x) is m,
and we write deg(f (x)) = m. The leading term of f (x) is a,mx", and the leading
coefficient is a,m. Note that the zero polynomial, 0, has no degree, leading term or
leading coefficient. A constant polynomial has degree 0 (or is the zero polynomial).
If R has an identity, then f (x) is monic if its leading coefficient is 1.
= 3+ 7x – 15x? + 0x³ + 2x4 + 0x°. Then
Example 10.2. In Q[x], let ƒ(x)
deg(f (x)) = 4, the leading term is 2x* and the leading coefficient is 2. This poly-
-
nomial is not monic.
We wish to make R[x] into a ring, and so we need addition and multiplication
operations. These will be exactly the same as for real polynomials. Let f(x) =
ao + ajx + · ..+ a„x" and g(x) = bo + bịx + ...+ bmx". Adding in terms with
zero coefficients if necessary, we may assume that m
= n. Then
f (x) + g(x) = (ao + bo) + (ai + bj)x + · . - + (a, + b„)x".
Similarly,
m+n
f (x)g(x) = co + c1x + c2x² +… .+ cm+nx'
where
C; = aob; + ab¡-1+a2b¡-2 +…+a;bo.
Here, we take a ; = 0 if j > n and b; = 0 if j > m.
= 5+2x+6x² and g (x) = 3+x+4x²+5x³. Then
Example 10.3. In Z7[x], let ƒ (x)
f (x)+g(x) = 1+3x+3x²+5x³ and ƒ (x)g(x) = 1+4x+5x² +4x³+6x4+2xS.
Theorem 10.1. If R is a ring, then so is R[x].
Corollary 10.1. Let R be a ring. Then
1. if R has an identity, then so does R[x]; and
2. if R is commutative, then so is R[x].
Theorem 10.2. Let R be an integral domain, and let f(x) and g(x) be nonzero
polynomials in R[x], of degree m and n respectively. Then
1. deg(f (x) + g(x)) is at most the larger of m and n (or f (x)+g(x) = 0); and
2. deg(ƒ(x)g(x)) :
— т + п.
Corollary 10.2. If R is an integral domain, then so is R[x].
Theorem 10.3. (Division Algorithm for Polynomials). Let F be a field, and let
f (x), g(x) E F[x], with g(x) # 0. Then there exist unique q(x), r(x) E F[x] such
that
f (x) = g(x)q(x)+r(x),
with either r(x) = 0 or deg(r(x)) < deg(g(x)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F73819141-031f-4c04-9ee3-6e038035be84%2Fws9x5yr_processed.png&w=3840&q=75)
Transcribed Image Text:Definition 10.1. Let R be a ring. Then a polynomial with coefficients in R is a
formal expression
ao + ajx + a2x² + · · · + anx",
where a; e Rand n is a nonnegative integer. Suppose that bo +bjx + • ·.+ bmx™
is also a polynomial with coefficients in R. Without loss of generality, let us say that
n < m. Then these polynomials are equal if and only if a; = b; for all i <n and
b; = 0 for alli > n. The set of all polynomials with coefficients in R is denoted
R[x].
Example 10.1. Let R = Z5. Then (inserting congruence class brackets for clarity),
an example of a polynomial in R[x] would be f (x) = [3] + [2]x + [4]x². As part
of the above definition, we observe that f (x) = g(x), where g(x) = [3] + [2]x +
[4]x? + [0]x³.
= ao + a¡x + · · · + a,,x" e R[x].
Definition 10.2. Let R be a ring and let f(x)
Further suppose that am +0 but ag = 0 for all k > m. Then the degree of f (x) is m,
and we write deg(f (x)) = m. The leading term of f (x) is a,mx", and the leading
coefficient is a,m. Note that the zero polynomial, 0, has no degree, leading term or
leading coefficient. A constant polynomial has degree 0 (or is the zero polynomial).
If R has an identity, then f (x) is monic if its leading coefficient is 1.
= 3+ 7x – 15x? + 0x³ + 2x4 + 0x°. Then
Example 10.2. In Q[x], let ƒ(x)
deg(f (x)) = 4, the leading term is 2x* and the leading coefficient is 2. This poly-
-
nomial is not monic.
We wish to make R[x] into a ring, and so we need addition and multiplication
operations. These will be exactly the same as for real polynomials. Let f(x) =
ao + ajx + · ..+ a„x" and g(x) = bo + bịx + ...+ bmx". Adding in terms with
zero coefficients if necessary, we may assume that m
= n. Then
f (x) + g(x) = (ao + bo) + (ai + bj)x + · . - + (a, + b„)x".
Similarly,
m+n
f (x)g(x) = co + c1x + c2x² +… .+ cm+nx'
where
C; = aob; + ab¡-1+a2b¡-2 +…+a;bo.
Here, we take a ; = 0 if j > n and b; = 0 if j > m.
= 5+2x+6x² and g (x) = 3+x+4x²+5x³. Then
Example 10.3. In Z7[x], let ƒ (x)
f (x)+g(x) = 1+3x+3x²+5x³ and ƒ (x)g(x) = 1+4x+5x² +4x³+6x4+2xS.
Theorem 10.1. If R is a ring, then so is R[x].
Corollary 10.1. Let R be a ring. Then
1. if R has an identity, then so does R[x]; and
2. if R is commutative, then so is R[x].
Theorem 10.2. Let R be an integral domain, and let f(x) and g(x) be nonzero
polynomials in R[x], of degree m and n respectively. Then
1. deg(f (x) + g(x)) is at most the larger of m and n (or f (x)+g(x) = 0); and
2. deg(ƒ(x)g(x)) :
— т + п.
Corollary 10.2. If R is an integral domain, then so is R[x].
Theorem 10.3. (Division Algorithm for Polynomials). Let F be a field, and let
f (x), g(x) E F[x], with g(x) # 0. Then there exist unique q(x), r(x) E F[x] such
that
f (x) = g(x)q(x)+r(x),
with either r(x) = 0 or deg(r(x)) < deg(g(x)).
![= 3x° + x+ + x³ + 3x² + 2x + 4 and g(x) = 2x³ + 3x² + x + 1
10.2. Let f (x)
be polynomials in Z5[x]. Find q(x), r (x) e Zs[x], with deg(r(x)) < 3, such that
f (x) = g(x)q(x) +r(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F73819141-031f-4c04-9ee3-6e038035be84%2Fioysle_processed.png&w=3840&q=75)
Transcribed Image Text:= 3x° + x+ + x³ + 3x² + 2x + 4 and g(x) = 2x³ + 3x² + x + 1
10.2. Let f (x)
be polynomials in Z5[x]. Find q(x), r (x) e Zs[x], with deg(r(x)) < 3, such that
f (x) = g(x)q(x) +r(x).
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