{"id":220585,"date":"2025-05-28T13:24:15","date_gmt":"2025-05-28T13:24:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=220585"},"modified":"2025-05-28T13:24:17","modified_gmt":"2025-05-28T13:24:17","slug":"please-tell-me-the-polynomials-and-why-are-they-polynomials-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/28\/please-tell-me-the-polynomials-and-why-are-they-polynomials-3\/","title":{"rendered":"Please tell me the polynomials and why are they polynomials"},"content":{"rendered":"\n

Please tell me the polynomials and why are they polynomials. Thank you for your time.<\/p>\n\n\n\n

Which of the following are polynomial functions?<\/p>\n\n\n\n

f(x) = 0
f(x) = -9
f(x) = 3 + 1
f(x) = 1\/2 + 8
f(x) = 32 + 2
f(x) = 45\/84
Check work<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

A polynomial function<\/strong> is a function of the form:<\/h3>\n\n\n\n

f(x)=anxn+an\u22121xn\u22121+\u22ef+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \\dots + a_1x + a_0<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • All exponents of xx are whole numbers (0, 1, 2, 3,…)<\/strong><\/li>\n\n\n\n
  • All coefficients aia_i are real numbers<\/strong><\/li>\n\n\n\n
  • No negative or fractional exponents<\/li>\n\n\n\n
  • No variables in denominators or under roots<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    \u2705 Polynomial Functions:<\/h3>\n\n\n\n
      \n
    1. f(x)=0f(x) = 0<\/strong>\n
        \n
      • This is the zero polynomial. All coefficients are zero, and it is valid.<\/li>\n\n\n\n
      • \u2705 Polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • f(x)=\u22129f(x) = -9<\/strong>\n
          \n
        • Constant function, degree 0.<\/li>\n\n\n\n
        • \u2705 Polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • f(x)=3x+1f(x) = 3x + 1<\/strong>\n
            \n
          • Degree 1 polynomial.<\/li>\n\n\n\n
          • \u2705 Polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
          • f(x)=x7\u221232×6\u2212\u03c0x3+4584f(x) = x^7 – 32x^6 – \\pi x^3 + \\frac{45}{84}<\/strong>\n
              \n
            • All exponents are whole numbers; constants (like \u03c0\\pi and fractions) are allowed as coefficients.<\/li>\n\n\n\n
            • \u2705 Polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
              \n\n\n\n

              \u274c Not Polynomial Functions:<\/h3>\n\n\n\n
                \n
              1. f(x)=x1\/2\u2212x+8f(x) = x^{1\/2} – x + 8<\/strong>\n
                  \n
                • Contains a fractional exponent x1\/2x^{1\/2} \u2192 not allowed<\/strong>.<\/li>\n\n\n\n
                • \u274c Not a polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
                • f(x)=\u22124x\u22123+5x\u22121+7\u221218x2f(x) = -4x^{-3} + 5x^{-1} + 7 – 18x^2<\/strong>\n
                    \n
                  • Contains negative exponents \u2192 not a polynomial<\/strong>.<\/li>\n\n\n\n
                  • \u274c Not a polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
                  • f(x)=(x+1)(x\u22121)+ex\u2212exf(x) = (x+1)(x-1) + e^x – e^x<\/strong>\n
                      \n
                    • ex\u2212ex=0e^x – e^x = 0, so it simplifies to (x+1)(x\u22121)=x2\u22121(x+1)(x-1) = x^2 – 1 \u2192 Polynomial.<\/li>\n\n\n\n
                    • \u2705 Polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
                    • f(x)=x2\u22123x+2x\u22122f(x) = \\frac{x^2 – 3x + 2}{x – 2}<\/strong>\n
                        \n
                      • This is a rational function<\/strong>, division by a variable expression \u2192 not a polynomial<\/strong><\/li>\n\n\n\n
                      • \u274c Not a polynomial<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
                        \n\n\n\n

                        \u2705 Final Answer (Polynomial functions):<\/h3>\n\n\n\n
                          \n
                        • f(x)=0f(x) = 0<\/li>\n\n\n\n
                        • f(x)=\u22129f(x) = -9<\/li>\n\n\n\n
                        • f(x)=3x+1f(x) = 3x + 1<\/li>\n\n\n\n
                        • f(x)=(x+1)(x\u22121)+ex\u2212exf(x) = (x+1)(x-1) + e^x – e^x<\/li>\n\n\n\n
                        • f(x)=x7\u221232×6\u2212\u03c0x3+4584f(x) = x^7 – 32x^6 – \\pi x^3 + \\frac{45}{84}<\/li>\n<\/ul>\n\n\n\n
                          \n\n\n\n

                          Explanation (300 words):<\/h3>\n\n\n\n

                          Polynomial functions are algebraic expressions consisting of variables and constants combined using only addition, subtraction, multiplication, and non-negative whole number exponents. They are defined over all real numbers and do not involve operations such as division by variables, roots (like square roots), or exponential functions like exe^x.<\/p>\n\n\n\n

                          To determine whether a function is a polynomial, we examine the powers of the variable and the nature of the coefficients. A polynomial must not include variables in the denominator (which creates a rational function), fractional or negative exponents (which imply roots or reciprocals), or transcendental functions like exponentials or logarithms.<\/p>\n\n\n\n

                          In the image, the functions f(x)=0f(x) = 0, f(x)=\u22129f(x) = -9, and f(x)=3x+1f(x) = 3x + 1 are straightforward polynomials of degree 0 or 1. The function f(x)=(x+1)(x\u22121)+ex\u2212exf(x) = (x+1)(x-1) + e^x – e^x simplifies to x2\u22121x^2 – 1 after canceling the exponential terms, which is a valid polynomial. The last function f(x)=x7\u221232×6\u2212\u03c0x3+4584f(x) = x^7 – 32x^6 – \\pi x^3 + \\frac{45}{84} is also a polynomial, even though it includes irrational (\u03c0\\pi) and fractional coefficients, since these are allowed in polynomials.<\/p>\n\n\n\n

                          However, functions like x1\/2\u2212x+8x^{1\/2} – x + 8 and \u22124x\u22123+5x\u22121+7\u221218×2-4x^{-3} + 5x^{-1} + 7 – 18x^2 are not polynomials due to fractional and negative exponents, respectively. Likewise, x2\u22123x+2x\u22122\\frac{x^2 – 3x + 2}{x – 2} is a rational function because it involves division by a variable expression.<\/p>\n\n\n\n

                          Thus, identifying polynomials hinges on recognizing proper structure and allowable operations.<\/p>\n\n\n\n

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

                          Please tell me the polynomials and why are they polynomials. Thank you for your time. Which of the following are polynomial functions? f(x) = 0f(x) = -9f(x) = 3 + 1f(x) = 1\/2 + 8f(x) = 32 + 2f(x) = 45\/84Check work The Correct Answer and Explanation is: A polynomial function is a function of […]<\/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-220585","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220585","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=220585"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220585\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=220585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=220585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=220585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}