{"id":436473,"date":"2018-05-29T00:45:16","date_gmt":"2018-05-29T00:45:16","guid":{"rendered":"https:\/\/essaypaper.org\/check-out-this-example-of-polynomial-with-solution\/"},"modified":"2018-10-24T09:06:24","modified_gmt":"2018-10-24T09:06:24","slug":"check-out-this-example-of-polynomial-with-solution","status":"publish","type":"post","link":"https:\/\/www.benedictsol.com\/blogs\/check-out-this-example-of-polynomial-with-solution\/","title":{"rendered":"Check Out This Example of Polynomial With Solution"},"content":{"rendered":"
\n

Task:<\/b><\/p>\n

Check whether the polynomials p<\/span>1<\/span>(x)=x<\/span>2<\/span>+4x, p<\/span>2<\/span>(x)=1+x<\/span>3<\/span>+x<\/span>2<\/span>, p<\/span>3<\/span>(x)=x<\/span>3<\/span>+x, p<\/span>4<\/span>(x)=x<\/span>2<\/span>-5 form a basis in the space P<\/span>3<\/span>.<\/span><\/p>\n

Solution:<\/b><\/p>\n

The standard basis in the space P<\/span>3<\/span> look like {1, x, x<\/span>2<\/span>, x<\/span>3<\/span>}, and they have 4 basis elements. To prove that polynomials p<\/span>1<\/span>, p<\/span>2<\/span>, p<\/span>3<\/span>, p<\/span>4<\/span> form a basis in the space P<\/span>3<\/span> it is sufficient to show that they are linearly independent.<\/span><\/p>\n

Let\u2019s form a linear combination:<\/span><\/p>\n

C<\/i><\/b>1<\/b> p<\/i><\/b>1<\/b> (<\/b> x<\/i><\/b>)<\/b> + C<\/i><\/b>2<\/b> p<\/i><\/b>2<\/b> (<\/b> x<\/i><\/b>)<\/b> + C<\/i><\/b>3<\/b> p<\/i><\/b>3<\/b> (<\/b> x<\/i><\/b>)<\/b> + C<\/i><\/b>4<\/b> p<\/i><\/b>4<\/b> (<\/b> x<\/i><\/b>)<\/b> = C<\/i><\/b>1<\/b> \u00a0(x <\/i><\/b>2<\/b> +<\/i><\/b>4<\/b> x) + \u00a0C<\/i><\/b>2<\/b> \u00a0(<\/i><\/b>1<\/b> + \u00a0x <\/i><\/b>3<\/b> +x <\/i><\/b>2<\/b> ) + C<\/i><\/b>3<\/b> \u00a0(x <\/i><\/b>3<\/b> + x) + C<\/i><\/b>4<\/b> ( x <\/i><\/b>2<\/b> \u2014 <\/i><\/b>5) =<\/b> C<\/i><\/b>1<\/b> x <\/i><\/b>2<\/b> + C<\/i><\/b>1<\/b> 4<\/b> x \u00a0+ C<\/i><\/b>2<\/b> \u00a0+ C<\/i><\/b>2<\/b> x <\/i><\/b>3<\/b> + C<\/i><\/b>2<\/b> x <\/i><\/b>2<\/b> \u00a0+ C<\/i><\/b>3<\/b> x <\/i><\/b>3<\/b> + C<\/i><\/b>3<\/b> x + C<\/i><\/b>4<\/b> x <\/i><\/b>2<\/b> \u00a0\u2013 C<\/i><\/b>4<\/b> 5 = <\/b><\/p>\n

x <\/i><\/b>3<\/b> (C<\/i><\/b>2<\/b> + C<\/i><\/b>3<\/b> ) + x <\/i><\/b>2<\/b> \u00a0(C<\/i><\/b>1<\/b> + C<\/i><\/b>2<\/b> + C<\/i><\/b>4<\/b> ) + \u00a0x (<\/i><\/b>4<\/b>C<\/i><\/b>1<\/b> + \u00a0C<\/i><\/b>3 <\/b>\u00a0<\/b> +<\/i><\/b> \u00a0<\/b>C<\/i><\/b>2<\/b> \u2013 <\/i><\/b>5<\/b>C<\/i><\/b>4 )<\/b><\/p>\n

This linear combination is equal to zero only if all coefficients are zero at powers of x, so we have come to the system:<\/span><\/p>\n

\"\"<\/p>\n

The system\u2019s only solution is C<\/span>1<\/span> = C<\/span>2=<\/span>C<\/span>3<\/span> = C<\/span>4<\/span> = 0 which means that the polynomials \u00a0p<\/span>1<\/span>, p<\/span>2<\/span>, p<\/span>3<\/span>, p<\/span>4 <\/span>are linearly independent and therefore form a basis in the space P<\/span>3<\/span>.<\/span><\/p>\n

\u00a0<\/p>\n

\n

In this <\/span><\/i>example of polynomial with solution<\/i>, you can find the answer to the question that was set by our professor. We want to inform you that Assignment.EssayShark.com can help you with your practical tasks. Since each student faces difficulties while studying from time to time, we have made it so that anyone can get help from our experts. There is a very successful solution to your problem with homework \u2013 place an order on our site. Tell us what your requirements are and don\u2019t forget to set the deadline while placing the order.<\/span><\/i><\/p>\n

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\u00a0<\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"

Task: Check whether the polynomials p1(x)=x2+4x, p2(x)=1+x3+x2, p3(x)=x3+x, p4(x)=x2-5 form a basis in the space P3. Solution: The standard basis in the space P3 look like {1, x, x2, x3}, and they have 4 basis elements. To prove that polynomials Read More …<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15,25],"tags":[],"class_list":["post-436473","post","type-post","status-publish","format-standard","hentry","category-essay-paper-writing","category-samples"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/436473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/comments?post=436473"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/436473\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=436473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=436473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=436473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}