{"id":1367,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1367"},"modified":"2015-11-12T18:35:29","modified_gmt":"2015-11-12T18:35:29","slug":"use-polynomial-division-to-solve-application-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/use-polynomial-division-to-solve-application-problems\/","title":{"raw":"Use polynomial division to solve application problems","rendered":"Use polynomial division to solve application problems"},"content":{"raw":"

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\n\n

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Example 6: Using Polynomial Division in an Application Problem<\/h3>\n

The volume of a rectangular solid is given by the polynomial [latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x\\\\[\/latex].\u00a0The length of the solid is given by 3x<\/em>\u00a0and the width is given by x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.<\/p>\n\n<\/div>\n

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Solution<\/h3>\n

There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 3<\/b>[\/caption]\n

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\n\n

[latex]\\begin{cases}V=l\\cdot w\\cdot h\\\\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h\\end{cases}\\\\[\/latex]<\/div>\n

To solve for h<\/em>, first divide both sides by 3x<\/em>.<\/p>\n\n

[latex]\\begin{cases}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\\\ \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{cases}\\\\[\/latex]<\/div>\n

Now solve for h<\/em>\u00a0using synthetic division.<\/p>\n\n

[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}\\\\[\/latex]<\/div>\n
\n\"Synthetic<\/a>\n

The quotient is [latex]{x}^{2}+x - 9\\\\[\/latex]\u00a0and the remainder is 0. The height of the solid is [latex]{x}^{2}+x - 9\\\\[\/latex].<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n

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Try It 3<\/h3>\n

The area of a rectangle is given by [latex]3{x}^{3}+14{x}^{2}-23x+6\\\\[\/latex].\u00a0The width of the rectangle is given by x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.<\/p>\nSolution<\/a>\n\n<\/div>","rendered":"

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\n

\n
\n
\n

Example 6: Using Polynomial Division in an Application Problem<\/h3>\n

The volume of a rectangular solid is given by the polynomial [latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x\\\\[\/latex].\u00a0The length of the solid is given by 3x<\/em>\u00a0and the width is given by x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.<\/p>\n<\/div>\n

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Solution<\/h3>\n

There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch.<\/p>\n

\"Graph<\/p>\n

Figure 3<\/b><\/p>\n<\/div>\n

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\n

[latex]\\begin{cases}V=l\\cdot w\\cdot h\\\\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h\\end{cases}\\\\[\/latex]<\/div>\n

To solve for h<\/em>, first divide both sides by 3x<\/em>.<\/p>\n

[latex]\\begin{cases}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\\\ \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{cases}\\\\[\/latex]<\/div>\n

Now solve for h<\/em>\u00a0using synthetic division.<\/p>\n

[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}\\\\[\/latex]<\/div>\n
\n\"Synthetic<\/a><\/p>\n

The quotient is [latex]{x}^{2}+x - 9\\\\[\/latex]\u00a0and the remainder is 0. The height of the solid is [latex]{x}^{2}+x - 9\\\\[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Try It 3<\/h3>\n

The area of a rectangle is given by [latex]3{x}^{3}+14{x}^{2}-23x+6\\\\[\/latex].\u00a0The width of the rectangle is given by x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.<\/p>\n

Solution<\/a><\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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