Difference between revisions of "Remainder and Factor Theorem"

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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>.
 
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>.
  
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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>.
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===Remainder Theorem===
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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 d===ivided 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>.
 
: 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>.
  
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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>.
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===Factor Theorem===
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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>.
 
: 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>.
  

Revision as of 11:20, 4 October 2021

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

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