{"id":227814,"date":"2025-06-06T13:58:27","date_gmt":"2025-06-06T13:58:27","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=227814"},"modified":"2025-06-06T13:58:28","modified_gmt":"2025-06-06T13:58:28","slug":"which-algebraic-expressions-are-polynomials","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/06\/which-algebraic-expressions-are-polynomials\/","title":{"rendered":"Which algebraic expressions are polynomials"},"content":{"rendered":"\n
Which algebraic expressions are polynomials? Check all that apply. 2x^3 - x^3y - 3x^2 + 6xy^2 + 5y - 2 - -x - x^3 - x^2 +<\/pre>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Definition of a Polynomial<\/strong><\/h3>\n\n\n\n
A polynomial<\/strong> is an algebraic expression that consists of variables and coefficients, combined using only addition, subtraction, and multiplication<\/strong>, and non-negative integer exponents<\/strong>.<\/p>\n\n\n\n
That means:<\/p>\n\n\n\n
    \n
  • No variable in the denominator<\/li>\n\n\n\n
  • No negative or fractional exponents<\/li>\n\n\n\n
  • No square roots or other radicals with variables<\/li>\n\n\n\n
  • Each term must be a product of a constant and variables raised to whole number exponents<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n
    \ud83e\uddea Let\u2019s evaluate the given expressions:<\/h3>\n\n\n\n
    1. 2×3\u2212x3y\u22123×2+6xy2+5y\u221222x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 22×3\u2212x3y\u22123×2+6xy2+5y\u22122<\/strong><\/h4>\n\n\n\n
      \n
    • All terms involve variables raised to non-negative integer exponents<\/strong><\/li>\n\n\n\n
    • All operations are addition\/subtraction\/multiplication<\/strong><\/li>\n\n\n\n
    • \u2705 This is a polynomial<\/strong><\/li>\n<\/ul>\n\n\n\n
      2. \u2212x\u2212x3\u2212x2-x – x^3 – x^2\u2212x\u2212x3\u2212x2<\/strong><\/h4>\n\n\n\n
        \n
      • Variables are raised to powers 1, 3, and 2<\/li>\n\n\n\n
      • No fractions, radicals, or negative exponents<\/li>\n\n\n\n
      • \u2705 This is a polynomial<\/strong><\/li>\n<\/ul>\n\n\n\n
        (If there were more options listed, we would evaluate those as well.)<\/h4>\n\n\n\n
        \n\n\n\n
        \u2705 Correct Answers<\/strong>:<\/h3>\n\n\n\n
          \n
        • 2×3\u2212x3y\u22123×2+6xy2+5y\u221222x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 22×3\u2212x3y\u22123×2+6xy2+5y\u22122<\/strong><\/li>\n\n\n\n
        • \u2212x\u2212x3\u2212x2-x – x^3 – x^2\u2212x\u2212x3\u2212x2<\/strong><\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n
          \ud83e\udde0 Explanation <\/h3>\n\n\n\n
          Polynomials are one of the foundational concepts in algebra. They are built from variables, constants (numbers), and exponents that are non-negative whole numbers. The structure of a polynomial is simple: it is a sum of terms<\/strong>, each of which is the product of a constant and one or more variables raised to whole-number powers.<\/p>\n\n\n\n
          For example, in the expression 2×3\u2212x3y\u22123×2+6xy2+5y\u221222x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 22×3\u2212x3y\u22123×2+6xy2+5y\u22122, we observe that each term:<\/p>\n\n\n\n
            \n
          • Has variables like xxx and yyy<\/li>\n\n\n\n
          • The powers on those variables are 0 or positive integers (like 3, 2, 1)<\/li>\n\n\n\n
          • There are no square roots, division by a variable, or variables with negative or fractional exponents<\/li>\n<\/ul>\n\n\n\n
            So, this expression qualifies as a multivariable polynomial<\/strong>.<\/p>\n\n\n\n
            The second expression, \u2212x\u2212x3\u2212x2-x – x^3 – x^2\u2212x\u2212x3\u2212x2, is also a polynomial. Though it has negative coefficients, this is allowed. What matters is that the exponents on the variables are all non-negative integers. This expression is a single-variable polynomial<\/strong> in xxx.<\/p>\n\n\n\n
            If an expression included, for example, terms like x\u22121x^{-1}x\u22121, 1\/x1\/x1\/x, x\\sqrt{x}x\u200b, or x1\/2x^{1\/2}x1\/2, it would not<\/strong> be a polynomial, because those involve negative or fractional exponents or roots.<\/p>\n\n\n\n
            In conclusion, polynomials are defined by the form of their terms<\/strong>, and both expressions listed meet that definition. Thus, they are polynomials.<\/p>\n\n\n\n
            \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
            Which algebraic expressions are polynomials? Check all that apply. 2x^3 – x^3y – 3x^2 + 6xy^2 + 5y – 2 – -x – x^3 – x^2 + The Correct Answer and Explanation is: Definition of a Polynomial A polynomial is an algebraic expression that consists of variables and coefficients, combined using only addition, subtraction, and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-227814","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227814","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=227814"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227814\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=227814"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=227814"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=227814"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}