{"id":1900,"date":"2024-06-24T19:07:01","date_gmt":"2024-06-24T19:07:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1900"},"modified":"2024-11-21T18:11:01","modified_gmt":"2024-11-21T18:11:01","slug":"introduction-to-power-and-polynomial-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-power-and-polynomial-functions-learn-it-3\/","title":{"raw":"Introduction to Power and Polynomial Functions: Learn It 3","rendered":"Introduction to Power and Polynomial Functions: Learn It 3"},"content":{"raw":"

Polynomial Functions<\/h2>\r\nNow that we've explored power functions and seen how a single term can have a powerful impact, let's take it a step further. Imagine combining multiple power functions into a single, more complex equation. This combination gives rise to polynomial functions, which are incredibly versatile and useful in modeling everything from the trajectory of a rocket to the fluctuations in the stock market.\r\n\r\n
An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently [latex]24[\/latex] miles in radius, but that radius is increasing by [latex]8[\/latex] miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex]\u00a0of the spill depends on the number of weeks [latex]w[\/latex]\u00a0that have passed. This relationship is linear.\r\n

[latex]r\\left(w\\right)=24+8w[\/latex]<\/p>\r\nWe can combine this with the formula for the area [latex]A[\/latex]\u00a0of a circle.\r\n

[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\r\nComposing these functions gives a formula for the area in terms of weeks.\r\n

[latex]\\begin{array}{l}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ A\\left(w\\right)=A\\left(24+8w\\right)\\\\ A\\left(w\\right)=\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\r\nMultiplying gives the formula below.\r\n

[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/p>\r\nThis formula is an example of a polynomial function<\/strong>.\r\n\r\n<\/section>Polynomial functions allow us to describe curves with multiple peaks and valleys, making them perfect for capturing the intricacies of real-world data.\r\n\r\n

\r\n

polynomial functions<\/h3>\r\nLet [latex]n[\/latex] be a non-negative integer. A polynomial function<\/strong> is a function that can be written in the form\r\n\r\n \r\n

[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\n \r\n\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<\/section>

Which of the following are polynomial functions?\r\n

[latex]\\begin{array}{c}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{array}[\/latex]<\/p>\r\n[reveal-answer q=\"906312\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"906312\"]\r\n\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

    \r\n \t
  • [latex]f\\left(x\\right)[\/latex] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\r\n \t
  • [latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\r\n \t
  • [latex]h\\left(x\\right)[\/latex] cannot be written in this form and is therefore not a polynomial function.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]24601[\/ohm2_question]<\/section>","rendered":"

    Polynomial Functions<\/h2>\n

    Now that we’ve explored power functions and seen how a single term can have a powerful impact, let’s take it a step further. Imagine combining multiple power functions into a single, more complex equation. This combination gives rise to polynomial functions, which are incredibly versatile and useful in modeling everything from the trajectory of a rocket to the fluctuations in the stock market.<\/p>\n

    An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently [latex]24[\/latex] miles in radius, but that radius is increasing by [latex]8[\/latex] miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex]\u00a0of the spill depends on the number of weeks [latex]w[\/latex]\u00a0that have passed. This relationship is linear.<\/p>\n

    [latex]r\\left(w\\right)=24+8w[\/latex]<\/p>\n

    We can combine this with the formula for the area [latex]A[\/latex]\u00a0of a circle.<\/p>\n

    [latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\n

    Composing these functions gives a formula for the area in terms of weeks.<\/p>\n

    [latex]\\begin{array}{l}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ A\\left(w\\right)=A\\left(24+8w\\right)\\\\ A\\left(w\\right)=\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\n

    Multiplying gives the formula below.<\/p>\n

    [latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/p>\n

    This formula is an example of a polynomial function<\/strong>.<\/p>\n<\/section>\n

    Polynomial functions allow us to describe curves with multiple peaks and valleys, making them perfect for capturing the intricacies of real-world data.<\/p>\n

    \n

    polynomial functions<\/h3>\n

    Let [latex]n[\/latex] be a non-negative integer. A polynomial function<\/strong> is a function that can be written in the form<\/p>\n

     <\/p>\n

    [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n

     <\/p>\n

    This 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>.<\/p>\n<\/section>\n

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

    [latex]\\begin{array}{c}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{array}[\/latex]<\/p>\n