Q4.4.4-2E ... [FREE SOLUTION]

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Chapter 4: Q4.4.4-2E (page 110)

Short Answer

Expert verified

If F is any field such that $f (x) \in F (x)$ and $k \in F$ . Then, k will be the root of the polynomial $f (x)$ , if and only if (x-k) is a factor of $f (x)$ in $F (x)$ .

Step by step solution

Factor Theorem:

If F is any field such that $f (x) \in F (x)$ and $k \in F$ . Then, k will be the root of the polynomial $f (x)$ , if and only if (x-k) is a factor of $f (x)$ in $F (x)$ .

Finding Remainder:

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Most popular questions from this chapter

Question: Let R be a commutative ring with identity and $a \in R$ . If $1_{R} + a x$ is a unit in $R [x]$ , show that $a^{n} = 0_{R}$ for some integer $n > 0$ . [Hint: Suppose that the inverse of $1_{R} + a x$ is $b_{0} + b_{1} x + b_{2} x^{2} + . . . + b_{k} x^{k}$ . Since their product is $1_{R}, b_{0} = 1_{R}$ (Why?) and the other coefficients are all $0_{R}$ .]

Let a be a fixed element of F and define a map $φ_{a} : F [x] \to F$ by $φ_{a} [f (x)] = f (a)$ . Prove that $φ_{a}$ is a surjective homomorphism of rings. The map $φ_{a}$ is called an evaluation homomorphism; there is one for each $a \in F$ .

Find the gcd of $x + a + b$ and $x^{3} - 3 a b x + a^{3} + b^{3}$ in $ℚ [x]$ .

If $f (x) \in F [x]$ , show that every non-zero constant polynomial divides role="math" localid="1648073927112" $f (x)$ .

Determine if the given polynomial is irreducible:

$x^{2} - 7 in ℝ [x]$ .
$x^{2} - 7 in ℚ [x]$ .
$x^{2} + 7 in ℂ [x]$ .
$2 x^{3} + x^{2} + 2 x + 2 in ℤ_{5} [x]$ .
$x^{3} - 9 in ℤ_{11} [x]$ .
role="math" localid="1648621615519" $x^{4} + x^{2} + 1 in ℤ_{3} [x]$ .

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Short Answer

Step by step solution

Factor Theorem:

Finding Remainder:

Finding Remainder:

Finding Remainder:

Finding Remainder:

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Most popular questions from this chapter

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