How can I add a custom “Text” widget to Appearance manager from functions.php?

I’m trying to add a replica of the default WP text widget, with my own css class parameter, to functions.php so that it appears in the widgets manager and can be added to a sidebar.

My first attempt is below, but I’m certain there has to be an easier way than how I’m doing it.

Can this be done in a simpler manner?

In functions.php, I’ve got this…

$google_search = TEMPLATEPATH . "/google_search.php";require_once($google_search);
add_action('widgets_init', create_function('', "register_widget('My_Widget_Search');"));

In google_search.php, I’ve got…
(Everything works except the textarea field contents aren’t being saved)

<?php

class My_Widget_Search extends WP_Widget {

    function My_Widget_Search() {
        $widget_ops = array( 'classname' => 'widget_search', 'description' => __( "Google Adsense Search Widget Placeholder" ) );
        $this->WP_Widget('adsense_search', __('Adsense Search Widget'), $widget_ops);
    }

    function widget( $args, $instance ) {
        extract( $args );
        $title = apply_filters('widget_title', empty( $instance['title'] ) ? __( '' ) : $instance['title']);
        $text = apply_filters('widget_text', empty( $instance['text'] ) ? __( '' ) : $instance['text']);

        }

    function update( $new_instance, $old_instance ) {
        $instance = $old_instance;
        $instance['title'] = strip_tags($new_instance['title']);
        $instance['text'] = $new_instance['text'];

        return $instance;
    }

    function form( $instance ) {
        //Defaults
        $instance = wp_parse_args( (array) $instance, array( 'title' => '') );
        $title = esc_attr( $instance['title'] );
        $text = $instance['text'];
?>
        <p><label for="<?php echo $this->get_field_id('title'); ?>"><?php _e( 'Title:' ); ?></label>
        <input class="widefat" id="<?php echo $this->get_field_id('title'); ?>" name="<?php echo $this->get_field_name('title'); ?>" type="text" value="<?php echo $title; ?>" /></p>
        <textarea class="widefat" rows="16" cols="20" id="<?php echo $this->get_field_id('text'); ?>" name=""<?php echo $this->get_field_name('text'); ?>"><?php echo $text; ?></textarea>
<?php
    }

}