Remainder and Factor Theorem - Revision history

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Created page with "The polynomial division algorithm is as follows: suppose d(x) and p(x) are nonzero polynomials where the degree of p(x) is greater than or equal to the degree of d(x). Then th..."

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The polynomial division algorithm is as follows: suppose d(x) and p(x) are nonzero polynomials where the degree of p(x) is greater than or equal to the degree of d(x). Then there exist two unique polynomials, q(x) and r(x), such that p(x) = d(x)q(x) + r(x), where either r(x) = 0 or the degree of r(x) is strictly less than the degree of d(x).

Remainder Theorem: Suppose p(x) is a polynomial of degree at least 1 and c is a real number. When p(x) is divided by x − c the remainder is p(c).
: Proof: By the division algorithm, p(x) = (x - c)q(x) + r, where r must be a constant since d(x) = x - c has a degree of 1. p(x) = (x - c)q(x) + r must hold for all values of x, so we can set x = c and get that p(c) = (c - c)q(x) + r = r. Thus the remainder r = p(c).

Factor Theorem: Suppose p(x) is a nonzero polynomial. The real number c is a zero of p if and only if (x − c) is a factor of p(x).
: By the division algorithm, x - c is a factor of p(x) if and only if r = 0. So, since p(c) = r when p(x) is divided by x - c, x - c is a factor of p(x) if and only if p(c) = 0; that is, if c is a zero of p(x).

==Resources==
* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem], Mathematics LibreTexts

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-The polynomial division algorithm is as follows: suppose <math> d(x) </math> and <math> p(x) </math> are nonzero polynomials where the degree of <math> p(x) </math> is greater than or equal to the degree of <math> d(x) </math>. Then there exist two unique polynomials, <math> q(x) </math> and <math> r(x) </math>, such that  <math> p(x) = d(x)q(x) + r(x) </math>, where either <math> r(x) = 0 </math> or the degree of <math> r(x) </math> is strictly less than the degree of <math> d(x) </math>.
+=Remainder Theorem=
-===Remainder Theorem===
+In algebra, the '''polynomial remainder theorem''' or '''little Bézout's theorem''' (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial <math>f(x)</math> by a linear polynomial <math>x-r</math> is equal to <math>f(r) .</math> In particular, <math>x-r</math> is a divisor of <math>f(x)</math> if and only if <math>f(r)=0,</math> a property known as the factor theorem.
-===Factor Theorem===
+== Examples ==
-Suppose <math> p(x) </math> is a nonzero polynomial. The real number <math> c </math> is a zero of <math> p(x) </math> if and only if <math> (x - c) </math> is a factor of <math> p(x) </math>.
+=== Example 1 ===
-: By the division algorithm, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> r = 0 </math>. So, since <math> p(c) = r </math> when <math> p(x) </math> is divided by <math> x - c </math>, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> p(c) = 0 </math>; that is, if <math> c </math> is a zero of <math> p(x) </math>.
+Let <math>f(x) = x^3 - 12x^2 - 42</math>. Polynomial division of <math>f(x)</math> by <math>(x-3)</math> gives the quotient <math>x^2 - 9x - 27</math> and the remainder <math>-123</math>. Therefore, <math>f(3)=-123</math>.
-==Resources==
+===Example 2===
 * [https://tutorial.math.lamar.edu/classes/alg/dividingpolynomials.aspx Dividing Polynomials], Paul's Online Notes
-== Licensing ==
+= Licensing =
 Content obtained and/or adapted from:
-* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem, Mathematics LibreTexts] under a CC BY-NC-SA license
+* [https://en.wikipedia.org/wiki/Polynomial_remainder_theorem Polynomial remainder theorem, Wikipedia] under a CC BY-SA license

← Older revision		Revision as of 22:41, 5 November 2021
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	==Resources==		==Resources==
−	* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem], Mathematics LibreTexts
	* [https://tutorial.math.lamar.edu/classes/alg/dividingpolynomials.aspx Dividing Polynomials], Paul's Online Notes		* [https://tutorial.math.lamar.edu/classes/alg/dividingpolynomials.aspx Dividing Polynomials], Paul's Online Notes
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem, Mathematics LibreTexts] under a CC BY-NC-SA license

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 The polynomial division algorithm is as follows: suppose <math> d(x) </math> and <math> p(x) </math> are nonzero polynomials where the degree of <math> p(x) </math> is greater than or equal to the degree of <math> d(x) </math>. Then there exist two unique polynomials, <math> q(x) </math> and <math> r(x) </math>, such that  <math> p(x) = d(x)q(x) + r(x) </math>, where either <math> r(x) = 0 </math> or the degree of <math> r(x) </math> is strictly less than the degree of <math> d(x) </math>.
-Remainder Theorem: Suppose <math> p(x) </math> is a polynomial of degree at least 1 and c is a real number. When <math> p(x) </math> is divided by <math> (x - c) </math> the remainder is <math> p(c) </math>.
+===Remainder Theorem===
 : Proof: By the division algorithm, <math> p(x) = (x - c)q(x) + r </math>, where r must be a constant since <math> d(x) = x - c </math> has a degree of 1. <math> p(x) = (x - c)q(x) + r </math> must hold for all values of <math> x </math>, so we can set <math> x = c </math> and get that <math> p(c) = (c - c)q(x) + r = r </math>. Thus the remainder <math> r = p(c) </math>.
-Factor Theorem: Suppose <math> p(x) </math> is a nonzero polynomial. The real number <math> c </math> is a zero of <math> p(x) </math> if and only if <math> (x - c) </math> is a factor of <math> p(x) </math>.
+===Factor Theorem===
 : By the division algorithm, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> r = 0 </math>. So, since <math> p(c) = r </math> when <math> p(x) </math> is divided by <math> x - c </math>, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> p(c) = 0 </math>; that is, if <math> c </math> is a zero of <math> p(x) </math>.

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 The polynomial division algorithm is as follows: suppose <math> d(x) </math> and <math> p(x) </math> are nonzero polynomials where the degree of <math> p(x) </math> is greater than or equal to the degree of <math> d(x) </math>. Then there exist two unique polynomials, <math> q(x) </math> and <math> r(x) </math>, such that  <math> p(x) = d(x)q(x) + r(x) </math>, where either <math> r(x) = 0 </math> or the degree of <math> r(x) </math> is strictly less than the degree of <math> d(x) </math>.
-Remainder Theorem: Suppose <math> p(x) </math> is a polynomial of degree at least 1 and c is a real number. When <math> p(x) </math> is divided by <math> x − c </math> the remainder is <math> p(c) </math>.
+Remainder Theorem: Suppose <math> p(x) </math> is a polynomial of degree at least 1 and c is a real number. When <math> p(x) </math> is divided by <math> (x - c) </math> the remainder is <math> p(c) </math>.
 : Proof: By the division algorithm, <math> p(x) = (x - c)q(x) + r </math>, where r must be a constant since <math> d(x) = x - c </math> has a degree of 1. <math> p(x) = (x - c)q(x) + r </math> must hold for all values of <math> x </math>, so we can set <math> x = c </math> and get that <math> p(c) = (c - c)q(x) + r = r </math>. Thus the remainder <math> r = p(c) </math>.
-Factor Theorem: Suppose <math> p(x) </math> is a nonzero polynomial. The real number <math> c </math> is a zero of <math> p(x) </math> if and only if <math> (x − c) </math> is a factor of <math> p(x) </math>.
+Factor Theorem: Suppose <math> p(x) </math> is a nonzero polynomial. The real number <math> c </math> is a zero of <math> p(x) </math> if and only if <math> (x - c) </math> is a factor of <math> p(x) </math>.
 : By the division algorithm, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> r = 0 </math>. So, since <math> p(c) = r </math> when <math> p(x) </math> is divided by <math> x - c </math>, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> p(c) = 0 </math>; that is, if <math> c </math> is a zero of <math> p(x) </math>.

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-The polynomial division algorithm is as follows: suppose d(x) and p(x) are nonzero polynomials where the degree of p(x) is greater than or equal to the degree of d(x). Then there exist two unique polynomials, q(x) and r(x), such that  <math> p(x) = d(x)q(x) + r(x) </math>, where either r(x) = 0 or the degree of r(x) is strictly less than the degree of d(x).
+The polynomial division algorithm is as follows: suppose <math> d(x) </math> and <math> p(x) </math> are nonzero polynomials where the degree of <math> p(x) </math> is greater than or equal to the degree of <math> d(x) </math>. Then there exist two unique polynomials, <math> q(x) </math> and <math> r(x) </math>, such that  <math> p(x) = d(x)q(x) + r(x) </math>, where either <math> r(x) = 0 </math> or the degree of <math> r(x) </math> is strictly less than the degree of <math> d(x) </math>.
-Remainder Theorem: Suppose p(x) is a polynomial of degree at least 1 and c is a real number. When p(x) is divided by x − c the remainder is p(c).
+Remainder Theorem: Suppose <math> p(x) </math> is a polynomial of degree at least 1 and c is a real number. When <math> p(x) </math> is divided by <math> x − c </math> the remainder is <math> p(c) </math>.
-: Proof: By the division algorithm, <math> p(x) = (x - c)q(x) + r </math>, where r must be a constant since d(x) = x - c has a degree of 1. p(x) = (x - c)q(x) + r must hold for all values of x, so we can set x = c and get that <math> p(c) = (c - c)q(x) + r = r </math>. Thus the remainder r = p(c).
+: Proof: By the division algorithm, <math> p(x) = (x - c)q(x) + r </math>, where r must be a constant since <math> d(x) = x - c </math> has a degree of 1. <math> p(x) = (x - c)q(x) + r </math> must hold for all values of <math> x </math>, so we can set <math> x = c </math> and get that <math> p(c) = (c - c)q(x) + r = r </math>. Thus the remainder <math> r = p(c) </math>.
-Factor Theorem: Suppose p(x) is a nonzero polynomial. The real number c is a zero of p if and only if (x − c) is a factor of p(x).
+Factor Theorem: Suppose <math> p(x) </math> is a nonzero polynomial. The real number <math> c </math> is a zero of <math> p(x) </math> if and only if <math> (x − c) </math> is a factor of <math> p(x) </math>.
-: By the division algorithm, x - c is a factor of p(x) if and only if r = 0. So, since p(c) = r when p(x) is divided by x - c, x - c is a factor of p(x) if and only if p(c) = 0; that is, if c is a zero of p(x).
+: By the division algorithm, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> r = 0 </math>. So, since <math> p(c) = r </math> when <math> p(x) </math> is divided by <math> x - c </math>, <math> x - c </math> is a factor of <math> p(x) </math> if and only if <math> p(c) = 0 </math>; that is, if <math> c </math> is a zero of <math> p(x) </math>.
 ==Resources==
 * [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem], Mathematics LibreTexts
 * [https://tutorial.math.lamar.edu/classes/alg/dividingpolynomials.aspx Dividing Polynomials], Paul's Online Notes

← Older revision		Revision as of 19:23, 22 September 2021
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	==Resources==		==Resources==
	* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem], Mathematics LibreTexts		* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(StitzZeager)/03%3A_Polynomial_Functions/3.02%3A_The_Factor_Theorem_and_the_Remainder_Theorem The Factor Theorem and Remainder Theorem], Mathematics LibreTexts
		+	* [https://tutorial.math.lamar.edu/classes/alg/dividingpolynomials.aspx Dividing Polynomials], Paul's Online Notes