Difference between revisions of "Remainder and Factor Theorem"
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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== | ==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 | * [https://tutorial.math.lamar.edu/classes/alg/dividingpolynomials.aspx Dividing Polynomials], Paul's Online Notes | ||
Revision as of 13:30, 22 September 2021
The polynomial division algorithm is as follows: suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x) } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } are nonzero polynomials where the degree of is greater than or equal to the degree of . Then there exist two unique polynomials, and , such that , where either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r(x) = 0 } or the degree of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r(x) } is strictly less than the degree of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x) } .
Remainder Theorem: Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } is a polynomial of degree at least 1 and c is a real number. When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } is divided by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x − c } the remainder is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(c) } .
- Proof: By the division algorithm, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = (x - c)q(x) + r } , where r must be a constant since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x) = x - c } has a degree of 1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = (x - c)q(x) + r } must hold for all values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } , so we can set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = c } and get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(c) = (c - c)q(x) + r = r } . Thus the remainder Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = p(c) } .
Factor Theorem: Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } is a nonzero polynomial. The real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c } is a zero of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x − c) } is a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } .
- By the division algorithm, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x - c } is a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 0 } . So, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(c) = r } when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } is divided by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x - c } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x - c } is a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(c) = 0 } ; that is, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c } is a zero of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } .
Resources
- The Factor Theorem and Remainder Theorem, Mathematics LibreTexts
- Dividing Polynomials, Paul's Online Notes
