{"id":1259,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1259"},"modified":"2017-03-31T17:47:12","modified_gmt":"2017-03-31T17:47:12","slug":"identify-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/identify-polynomial-functions\/","title":{"raw":"Identify polynomial functions","rendered":"Identify polynomial functions"},"content":{"raw":"An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r<\/em>\u00a0of the spill depends on the number of weeks w<\/em>\u00a0that have passed. This relationship is linear.\r\n[latex]\\left(w\\right)=24+8w[\/latex]\r\nWe can combine this with the formula for the area A<\/em>\u00a0of a circle.\r\n[latex]\\left(w\\right)=\\pi {r}^{2}[\/latex]\r\nComposing these functions gives a formula for the area in terms of weeks.\r\n[latex]\\begin{cases}\\left(w\\right)=\\left(\\left(\\right)\\right)\\\\ =\\left(24+8w\\right)\\\\ =\\pi {\\left(24+8w\\right)}^{2}\\end{cases}[\/latex]\r\nMultiplying gives the formula.\r\n[latex]\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]\r\nThis formula is an example of a polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.\r\n
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A General Note: Polynomial Functions<\/h3>\r\nLet n<\/em>\u00a0be a non-negative integer. A polynomial function<\/strong> is a function that can be written in the form\r\n[latex]f\\left(\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]\r\nThis is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a term of a polynomial function<\/strong>.\r\n\r\n<\/div>\r\n
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Example 4: Identifying Polynomial Functions<\/h3>\r\nWhich of the following are polynomial functions?\r\n[latex]\\begin{cases}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{cases}[\/latex]\r\n
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Solution<\/h3>\r\nThe first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.\r\n