{"id":42176,"date":"2025-06-28T16:20:19","date_gmt":"2025-06-28T16:20:19","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=42176"},"modified":"2025-06-28T16:20:20","modified_gmt":"2025-06-28T16:20:20","slug":"factor-the-expression-over-the-complex-numbers","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/factor-the-expression-over-the-complex-numbers\/","title":{"rendered":"Factor the expression over the complex numbers."},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer:<\/strong> (y – 7i)(y + 7i)<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The task is to factor the expression y\u00b2 + 49 over the set of complex numbers. This expression is a specific type known as a “sum of squares”. While a sum of squares cannot be factored using only real numbers, it can be factored using complex numbers by applying the “difference of squares” formula.<\/p>\n\n\n\n

The difference of squares formula is a\u00b2 – b\u00b2 = (a – b)(a + b). Our goal is to manipulate y\u00b2 + 49 so it fits this pattern. To achieve this, we introduce the imaginary unit, i, which is defined by the fundamental property i\u00b2 = -1. This property allows us to convert a positive term into a negative one.<\/p>\n\n\n\n

First, we can rewrite the expression y\u00b2 + 49 as a subtraction:
y\u00b2 + 49 = y\u00b2 – (-49)<\/p>\n\n\n\n

Next, we need to find a term that, when squared, equals -49. Using the imaginary unit i, we can express -49 as the square of 7i:
(7i)\u00b2 = 7\u00b2 * i\u00b2 = 49 * (-1) = -49<\/p>\n\n\n\n

Now we can substitute (7i)\u00b2 back into our expression:
y\u00b2 – (-49) = y\u00b2 – (7i)\u00b2<\/p>\n\n\n\n

This expression is now in the form of a difference of squares, a\u00b2 – b\u00b2, where a = y and b = 7i. We can apply the factoring formula (a – b)(a + b):
y\u00b2 – (7i)\u00b2 = (y – 7i)(y + 7i)<\/p>\n\n\n\n

Therefore, the expression y\u00b2 + 49 factored over the complex numbers is (y – 7i)(y + 7i). We can verify this by multiplying the factors, which results in y\u00b2 + 7iy – 7iy – 49i\u00b2, simplifying to y\u00b2 + 49.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

The Correct Answer and Explanation is: Correct Answer: (y – 7i)(y + 7i) Explanation: The task is to factor the expression y\u00b2 + 49 over the set of complex numbers. This expression is a specific type known as a “sum of squares”. While a sum of squares cannot be factored using only real numbers, it can […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-42176","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42176","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=42176"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42176\/revisions"}],"predecessor-version":[{"id":42179,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42176\/revisions\/42179"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=42176"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=42176"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=42176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}