{"id":40257,"date":"2025-06-27T15:12:40","date_gmt":"2025-06-27T15:12:40","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=40257"},"modified":"2025-06-27T15:12:42","modified_gmt":"2025-06-27T15:12:42","slug":"use-synthetic-division-to-find-the-quotient-and-remainder-in-each-of-the-following","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/use-synthetic-division-to-find-the-quotient-and-remainder-in-each-of-the-following\/","title":{"rendered":"Use synthetic division to find the quotient and remainder in each of the following."},"content":{"rendered":"\n

Use synthetic division to find the quotient and remainder in each of the following. Write your complete solutions at the back of your module. 1. (3×3 + x2 – 22x – 25) – (x – 2) Quotient: Remainder: 2. (x3 + 4×2 – x – 25) = (x + 5) Quotient: Remainder: 3. (6×3 – 5×2 + 4x – 1) = (3x – 1) Quotient: Remainder: 4. (2×4 – 9×3 + 9×2 + 5x – 1) = (2x + 1) Quotient: Remainder: 5. (2×4 + 5×3 + 3×2 + 8x + 12) = (2x + 3) Quotient: Remainder:<\/p>\n\n\n\n

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

\n\n\n\n

1.<\/strong> Divide: (3x\u00b3 + x\u00b2 – 22x – 25) \u00f7 (x – 2)<\/em><\/h3>\n\n\n\n

Setup:<\/strong> Use synthetic division with root 2<\/strong><\/p>\n\n\n\n

luaCopyEdit2 |  3   1   -22   -25  \n   |      6   14   -16  \n   -------------------  \n      3   7   -8   -41  \n<\/code><\/pre>\n\n\n\n
Quotient:<\/strong> 3x\u00b2 + 7x – 8<\/em>
Remainder:<\/strong> -41<\/em><\/p>\n\n\n\n
\n\n\n\n
2.<\/strong> Divide: (x\u00b3 + 4x\u00b2 – x – 25) \u00f7 (x + 5)<\/em><\/h3>\n\n\n\n
Setup:<\/strong> Root is -5<\/strong><\/p>\n\n\n\n
diffCopyEdit-5 |  1   4   -1   -25  \n    |     -5   5   -20  \n    ------------------  \n       1  -1   4   -45  \n<\/code><\/pre>\n\n\n\n
Quotient:<\/strong> x\u00b2 – x + 4<\/em>
Remainder:<\/strong> -45<\/em><\/p>\n\n\n\n
\n\n\n\n
3.<\/strong> Divide: (6x\u00b3 – 5x\u00b2 + 4x – 1) \u00f7 (3x – 1)<\/em><\/h3>\n\n\n\n
Setup:<\/strong> Root is 1\/3<\/strong><\/p>\n\n\n\n
luaCopyEdit1\/3 |  6   -5   4   -1  \n     |     2  -1   1  \n     ----------------  \n         6  -3   3   0  \n<\/code><\/pre>\n\n\n\n
Quotient:<\/strong> 6x\u00b2 – 3x + 3<\/em>
Remainder:<\/strong> 0<\/em><\/p>\n\n\n\n
\n\n\n\n
4.<\/strong> Divide: (2x\u2074 – 9x\u00b3 + 9x\u00b2 + 5x – 1) \u00f7 (2x + 1)<\/em><\/h3>\n\n\n\n
Setup:<\/strong> Root is -1\/2<\/strong><\/p>\n\n\n\n
diffCopyEdit-1\/2 |  2   -9   9   5   -1  \n      |     -1   5  -7   1  \n      ---------------------  \n           2  -10  14  -2   0  \n<\/code><\/pre>\n\n\n\n
Quotient:<\/strong> 2x\u00b3 – 10x\u00b2 + 14x – 2<\/em>
Remainder:<\/strong> 0<\/em><\/p>\n\n\n\n
\n\n\n\n
5.<\/strong> Divide: (2x\u2074 + 5x\u00b3 + 3x\u00b2 + 8x + 12) \u00f7 (2x + 3)<\/em><\/h3>\n\n\n\n
Setup:<\/strong> Root is -3\/2<\/strong><\/p>\n\n\n\n
diffCopyEdit-3\/2 |  2   5   3   8   12  \n      |    -3   -3   0  -12  \n      ----------------------  \n          2   2   0   8   0  \n<\/code><\/pre>\n\n\n\n
Quotient:<\/strong> 2x\u00b3 + 2x\u00b2 + 0x + 8<\/em>
Remainder:<\/strong> 0<\/em><\/p>\n\n\n\n
\n\n\n\n
Explanation of Synthetic Division<\/strong><\/h2>\n\n\n\n
Synthetic division is a shortcut for dividing polynomials when the divisor is a linear factor in the form (x – k)<\/em> or (ax + b)<\/em>. It is faster and requires fewer steps compared to long division. You start by writing only the coefficients of the dividend and applying the root derived from the divisor. The root is the value that makes the divisor zero. For instance, for (x – 2)<\/em> the root is 2<\/strong>, for (x + 5)<\/em> the root is -5<\/strong>, and so on.<\/p>\n\n\n\n
In the synthetic setup, multiply the root by the current term, add the result to the next coefficient, and repeat until the remainder is found. The last number is always the remainder, while the other terms form the quotient polynomial. This method works efficiently with proper arrangement of terms in descending order and is useful in polynomial factorization or finding zeros.<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
Use synthetic division to find the quotient and remainder in each of the following. Write your complete solutions at the back of your module. 1. (3×3 + x2 – 22x – 25) – (x – 2) Quotient: Remainder: 2. (x3 + 4×2 – x – 25) = (x + 5) Quotient: Remainder: 3. (6×3 – […]<\/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-40257","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40257","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=40257"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40257\/revisions"}],"predecessor-version":[{"id":40259,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40257\/revisions\/40259"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=40257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=40257"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=40257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}