/images/image.jpg not working in sidebar

PHP code won’t run inside of a text widget. There are plugins that will let you do that, but this type of thing is strongly discouraged.

Add this code to your functions.php file, or better yet, make it a simple little plugin:

// Enable the use of shortcodes within widgets.
add_filter( 'widget_text', 'do_shortcode' ); 

// Assign the tag for our shortcode and identify the function that will run. 
add_shortcode( 'template_directory_uri', 'wpse61170_template_directory_uri' );

// Define function 
function wpse61170_template_directory_uri() {
    return get_template_directory_uri();
}

To use it in the admin, add the shortcode [template_directory_uri] to your text widget, and the template directory will be output e.g.

[template_directory_uri]/images/image.jpg