11 x1 t15 04 polynomial theorems (2012)

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<ul><li><p>Polynomial Theorems</p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x </p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x let ( ) ( )A x x a </p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x let ( ) ( )A x x a </p><p> ( ) ( ) ( )P x x a Q x R x </p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x let ( ) ( )A x x a </p><p> ( ) ( ) ( )P x x a Q x R x ( ) ( ) ( )P a a a Q a R a </p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x let ( ) ( )A x x a </p><p> ( ) ( ) ( )P x x a Q x R x ( ) ( ) ( )P a a a Q a R a </p><p>( )R a</p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x let ( ) ( )A x x a </p><p> ( ) ( ) ( )P x x a Q x R x ( ) ( ) ( )P a a a Q a R a </p><p>( )R anow degree ( ) 1</p><p>( ) is a constantR x</p><p>R x</p></li><li><p>Polynomial TheoremsRemainder TheoremIf the polynomial P(x) is divided by (x a), then the remainder is P(a)</p><p>Proof: ( ) ( ) ( )P x A x Q x R x let ( ) ( )A x x a </p><p> ( ) ( ) ( )P x x a Q x R x ( ) ( ) ( )P a a a Q a R a </p><p>( )R anow degree ( ) 1</p><p>( ) is a constantR x</p><p>R x</p><p>( ) ( )</p><p>( )R x R a</p><p>P a</p></li><li><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x </p></li><li><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x </p></li><li><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p></li><li><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 </p></li><li><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x22x</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x22x 19 30x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x 19 30x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x 19 30x 22 4x x</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x30</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x30</p><p>15</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x30</p><p>15</p><p>15 30x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x30</p><p>15</p><p>15 30x 0</p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x30</p><p>15</p><p>15 30x 0</p><p> 2( ) ( 2) 2 15P x x x x </p></li><li><p>Factor TheoremIf (x a) is a factor of P(x) then P(a) = 0</p><p>e.g. Find the remainder when is divided by (x 2)</p><p> 3 25 17 11P x x x x 3 25 17 11P x x x x 3 22 5 2 17 2 2 11P </p><p>19 remainder when ( ) is divided by ( 2) is 19P x x </p><p>e.g. (i) Show that (x 2) is a factor of and hence factorise P(x).</p><p> 3 19 30P x x x 32 2 19 2 30P </p><p>0( 2) is a factorx </p><p>3 22 0 19 30x x x x 2x</p><p>3 22x x2x</p><p>22x</p><p>15x19 30x </p><p>22 4x x30</p><p>15</p><p>15 30x 0</p><p> 2( ) ( 2) 2 15P x x x x ( 2)( 5)( 3)x x x </p></li><li><p>OR 3 19 30P x x x </p></li><li><p>OR 3 19 30P x x x ( 2) x </p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have?</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15x</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need?</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19x</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need?</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need? 4x</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need? 4x</p><p>4 2 ?x </p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need? 4x</p><p>4 2 ?x </p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need? 4x</p><p>4 2 ?x </p><p>2x</p><p> 2( ) ( 2) 2 15P x x x x ( 2)( 5)( 3)x x x </p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term leading term=leading term</p><p>2x</p><p>constant constant=constant</p><p>15</p><p>If you where to expand out now, how many x would you have? 15xHow many x do you need? 19xHow do you get from what you have to what you need? 4x</p><p>4 2 ?x </p><p>2x</p></li><li><p>OR 3 19 30P x x x ( 2) x </p><p>leading term...</p></li></ul>