In your theme, add this function, and replace mytheme with your theme name:
/**
* @param $form
* @param $form_state
* @param $form_id
*/
function mytheme_form_alter(&$form, $form_state, $form_id) {
switch ($form_id){
case 'search_form':
unset($form['help_link']);
unset($form['advanced']);
break;
}
}
