Word Problems - Polynomials
This lesson is meant to summarize all the major concepts covered from this unit.
There are a few different ways we can factor or find the factors of a function. We can use either Synthetic Division for dividing a polynomial by a linear expression or Long Division for dividing a polynomial by another polynomial. You can also use the Rational Root Theorem to determine the divisor for these calculations.
When determining Rates of Change, it's important to understand what you're calculating. If you're determining the rate of change over a given interval, use the \(A_{roc}\) formula. If you're calculating the rate of change at a single instance, use the \(I_{roc}\) formula.
There are two roller coasters travelling at the same time. The heights of the coasters are modelled during \(5\) seconds. The height of the first coaster in meters is modelled by \(h(t) = t^3 -8t^2 + 32t -40\). The height of the second coaster in meters is modelled by \(h(t) = -t^2 + 18t - 32\)
- Determine the real roots of the function for the height of the first coaster
- Determine the real roots of the function for the height of the second coaster
- At what times are the two coasters at the same height?
- What are the heights that correspond to the times in part iii?
i. We can determine the real roots of the function for the height of the first coaster by using the rational root theorem \(\left(\cfrac{p}{q}\right)\) where \(p\) represents the factor of the trailing constant whereas \(q\) represents the factor of the leading coefficient.
The factors of the leading coefficient, \(1\), is \(1\). The factors of the constant term, \(-8\) are \(1, 2, 4, 5, 8, 10, 20, 40\).
If we substitute these values into the current expression, we get \(P(2) = 0\).
\(h(2) = (2)^3 -8(2)^2 + 32(2) -40\)
\(h(2) = 8 -8(4) + 64 -40\)
\(h(2) = 8 -32 + 64 -40\)
\(h(2) = 0\)
Therefore, we can determine that the first coaster has a real root at \(\boldsymbol{t = 2}\).
ii. We can determine the real roots of the function for the height of the second coaster by setting it equal to \(0\) and factoring:
\(0 = -t^2 + 18t - 32\)
\(0 = (-t + 2)(t - 16)\)
\(t_1 = 2\)
\(t_2 = 16\)
Therefore, we can determine that the real roots of the function for the height of the second coaster are \(\boldsymbol{t = 2, 16}\).
iii. In order to determine what times the two coasters are at the same height, we can first set them equal to each other:
Next, we can move all terms onto one side and collect like terms:
\(t^3 -8t^2 + t^2 + 32t - 18t - 40 + 32 = 0\)
\(t^3 -7t^2 + 14t - 8 = 0\)
Then, we can factor the equation. To find the factors of this equation, we can use the rational root theorem \(\left(\cfrac{p}{q}\right)\) where \(p\) represents the factor of the trailing constant whereas \(q\) represents the factor of the leading coefficient.
The factors of the leading coefficient, \(1\), is \(1\). The factors of the constant term, \(-8\) are \(1, 2, 4, 8\).
If we substitute these values into the current expression, we get \(P(1) = 0\).
\(P(1) = (1)^3 -7(1)^2 + 14(1) - 8\)
\(P(1) = 0\)
After, we can use Synthetic Division to determine the additonal factors of the factored expression:
\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 1 & -7 & 14 & -8\\ {\color{red}{1}} & \downarrow & 1 & -6 & 8\\ \hline & 1 & -6 & 8 &|\phantom{-} {\color{blue}0} \end{array}Based on performing Synthetic Division, we can determine that the factors of the factored expression are \(1, -6,\) and \(8\). Therefore, we can write the factored expression as such:
We can factor this expression further to determine the times:
\(f(t) = (t-1)(t-2)(t-4)\)
\(t_1 = 1\)
\(t_2 = 2\)
\(t_3 = 4\)
Therefore, we can determine that the two roller coasters are at the same height at \(\boldsymbol{t = 1, 2, 4}\).
iv. In order to determine which heights correspond to the times in the last part, we can substitute them into one of the original equations. In this instance, we will substitute them into the first equation:
We can determine the height at \(t=1\) as such:
\(h(1) = (1)^3 -8(1)^2 + 32(1) -40\)
\(h(1) = -15\)
We can determine the height at \(t=2\) as such:
\(h(2) = (2)^3 -8(2)^2 + 32(2) -40\)
\(h(2) = 0\)
We can determine the height at \(t=4\) as such:
\(h(4) = (4)^3 -8(4)^2 + 32(4) -40\)
\(h(4) = 24\)
Therefore, we can determine that the heights corresponding to the times \(1, 2,\) and \(4\) are \(-15, 0,\) and \(24\) respectively. Since \(-15\) represents a negative height, this means that the 2 roller coasters only reach the same heights at \(\boldsymbol{2}\) and \(\boldsymbol{4}\) seconds.
A box has a length of \(2x^2 -1\) units a width of \(x+3\) units and a height of \(2\) units
- Write a function for the box in terms of \(x\)
- Determine the values of \(x\) that will produce a Volume of \(0\)
- What is the domain for this real-life situation?
- What are the dimensions of the box if the volume is \(204\;[\text{cm}^3]\)
i. First, we can rewrite the formula for the Volume of a rectangular prism:
Since we can determine that \(l = 2x^2 -1\), \(w = x+ 3\), and \(h = 2\), we can substitute these values into the formula:
\(V = (2x^2 -1)(x+ 3)(2)\)
\(V = 2(2x^2 -1)(x+ 3)\)
Therefore, we can determine that the function of the box expressed in terms of \(x\) is \(\boldsymbol{V = (2x^2 -1)(x+ 3)(2)}\).
ii. In order to determine the values of \(x\) that will produce a Volume of \(0\), we can set the function equal to \(0\):
Next, we can factor the function further and determine the roots:
\(0 = 4\left(x^2 -\cfrac{1}{2}\right)(x+ 3)\)
\(0 = 4\left(x +\cfrac{1}{\sqrt{2}}\right)\left(x -\cfrac{1}{\sqrt{2}}\right)(x+ 3)\)
\(x_1 = -3\)
\(x_2 = -\cfrac{1}{\sqrt{2}}\)
\(x_3 = \cfrac{1}{\sqrt{2}}\)
Therefore, we can determine that the values of \(x\) that will produce a Volume of \(0\) are \(\boldsymbol{x = -3, \pm \cfrac{1}{\sqrt{2}}}\).
iii. In order to determine the domain of this function, we can first sketch a graph:
Based on this graph, we can determine the domain as \(\boldsymbol{x \in \left(-3, -\cfrac{1}{\sqrt{2}}\right) \cup \left(\cfrac{1}{\sqrt{2}}, \infty \right)}\).
iv. In order to determine what the dimensions of the box will be if the Volume is \(204\;[cm^3]\), we can first set the function equal to \(204\):
Next, we can expand the function and move all terms onto one side:
\(204 = 2(2x^3 + 6x^2 - x - 3)\)
\(204 = 4x^3 + 12x^2 - 2x - 6\)
\(0 = 4x^3 + 12x^2 - 2x - 210\)
\(0 = 2(2x^3 + 6x^2 - x - 105)\)
Then, we can factor the function even further. In order to do so, we use the rational root theorem \(\left(\cfrac{p}{q}\right)\) where \(p\) represents the factor of the trailing constant while \(q\) represents the factor of the leading coefficient. The factors of the leading coefficient, \(2\), are \(1, 2\) while the factors of the constant term, \(-105\) are \(1, 3, 5, 7, 15, 21, 35, 105\).
If we substitute these values into the current expression, we get \(f(3) = 0\):
\(f(3) = 2[2(3)^3 + 6(3)^2 - 3 - 105]\)
\(f(3) = 0\)
After, we can use Synthetic Division to determine the additonal factors of the factored expression:
\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 2 & 6 & -1 & -105\\ {\color{red}{3}} & \downarrow & 6 & 36 & -135\\ \hline & 2 & 12 & 35 &|\phantom{-} {\color{blue}0} \end{array}Based on performing Synthetic Division, we can determine that the factors of the factored expression are \(2, 12\,\) and \(35\). Therefore, we can write the factored expression as such:
Finally, we can subsitute \(3\) into the equations of the original equations to determine their side lengths.
We can determine the length as such:
\(l = 2x^2 -1\)
\(l = 2(3)^2 - 1\)
\(l = 18 - 1\)
\(l = 17\;[\text{cm}]\)
We can determine the width as such:
\(w = x + 3\)
\(w = 3 + 3\)
\(w = 6 \; [\text{cm}]\)
We can determine the height as such:
\(h = 2\;[\text{cm}]\)
We can multiply these side lengths to verify that they result in a Volume of \(204\;[\text{cm}^3]\):
\(V = l \cdot w \cdot h\)
\(V = (17)(6)(2)\)
\(V = 204\;[\text{cm}^3]\)
Therefore, we can determine the dimensions of the box are \(\boldsymbol{17\;[\textbf{cm}]}\), \(\boldsymbol{6 \; [\textbf{cm}]}\), and \(\boldsymbol{2\;[\textbf{cm}]}\) if the Volume of the box is \(204\;[\text{cm}^3]\).
Sketch the polynomial \(f(x) = -3x^3 + 7x^2 + 22x -8\) by showing all the details. Follow the following steps:
- Find the \(y\)-intercept
- Factor the polynomial to find zeroes
- Determine the end behaviours
- Find the equation for the slope of the tangent at any point \(x\) (use the difference quotient)
- Find where the tangent slope is \(0\). This will indicate where the turning points are
- Find the \(y\)-coordinates of the turning points
- Sketch the turning points and connect all the sketched information with a smoothed curve
i. We can determine the \(y\)-intercept by setting \(x = 0\):
\(f(0) = -3(0)^3 + 7(0)^2 + 22(0) -8\)
\(f(0) = -8\)
Therefore, we can determine the \(y\)-intercept is \(\boldsymbol{(0, -8)}\).
ii. In order to factor the function, we can use the rational root theorem \(\left(\cfrac{p}{q}\right)\) where \(p\) represents the factor of the trailing constant while \(q\) represents the factor of the leading coefficient. The factors of the leading coefficient, \(-3\), are \(\pm 1, 3\) while the factors of the constant term, \(-8\) are \(\pm 1, 2, 4, 8\).
If we substitute these values into the current expression, we get \(f(-2) = 0\):
\(f(-2) = -3(-2)^3 + 7(-2)^2 + 22(-2) -8\)
\(f(-2) = -3(-8) + 7(4) - 52\)
\(f(-2) = 24 + 28 - 52\)
\(f(-2) = 0\)
After, we can use Synthetic Division to determine the additonal factors of the factored expression:
\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & -3 & 7 & 22 & -8\\ {\color{red}{-2}} & \downarrow & 6 & -26 & 8\\ \hline & -3 & 13 & -4 &|\phantom{-} {\color{blue}0} \end{array}Based on performing Synthetic Division, we can determine that the factors of the factored expression are \(-3, 13\,\) and \(-4\). Therefore, we can write the factored expression as such:
We can factor the expression further:
We can find the zeroes by setting the function equal to \(0\):
\(0 = (x + 2)(x-4)(-3x + 1)\)
\(x_1 = -2\)
\(x_2 = 4\)
\(x_3 = \cfrac{1}{3}\)
Therefore, we can determine that the zeroes are \(\boldsymbol{x = -2, \cfrac{1}{3}, 4}\).
iii. Given that we are looking at an odd polynomial function (degree \(3\)) with a negative leading coefficient \((-3)\), we can summarize the end behaviours as such:
\(x \rightarrow \infty, y \rightarrow -\infty\)
\(x \rightarrow -\infty, y \rightarrow +\infty\)
iv. We can use the Difference Quotient to determine the equation of the tangent's slope at any point \(x\):
\(m_{\tan} = \cfrac{[-3(x+h)^3 + 7(x+h)^2 + 22(x+h) -8] - [-3x^3 + 7x^2 + 22x -8]}{h}\)
\(m_{\tan} = \cfrac{-3(x^3 + 3x^2h + 3xh^2 + h^3) + 7(x^2 + 2xh + h^2) + \cancel{22x} + 22h \cancel{-8} + 3x^3 - 7x^2 \cancel{-22x} +\cancel{8}}{h}\)
\(m_{\tan} = \cfrac{\cancel{-3x^3} - 9x^2h - 9xh^2 - 3h^3 + \cancel{7x^2} + 14xh + 7h^2 + 22h + \cancel{3x^3} \cancel{-7x^2}}{h}\)
\(m_{\tan} = \cfrac{\cancel{h}(-9x^2 - 9xh - 3h^2 + 14x + 7h + 22)}{\cancel{h}}\)
\(m_{\tan} = -9x^2 - 9xh - 3h^2 + 14x + 7h + 22\)
Next, we can set \(h = 0\) to get the equation of the slope's tangent:
\(m_{\tan} = -9x^2 - 9x(0) - 3(0)^2 + 14x + 7(0) + 22\)
\(m_{\tan} = -9x^2 + 14x + 22\)
Therefore, we can determine that the equation for the slope of the tangent at any point \(x\) is \(\boldsymbol{m_{\tan} = -9x^2 + 14x + 22}\).
v. First, we can determine where the tangent slope is \(0\) by setting the equation equal to \(0\):
Next, we can use the quadratic formula to determine the roots. We can determine that \(\textcolor{red}{a = -9}\), \(\textcolor{green}{b = 14}\), and \(\textcolor{blue}{c = 22}\):
\(x = \cfrac{-\textcolor{green}{14} \pm \sqrt{(\textcolor{green}{14})^2 - 4(\textcolor{red}{-9})(\textcolor{blue}{22})}}{2(\textcolor{red}{-9})}\)
\(x = \cfrac{-14 \pm \sqrt{196 + 792}}{-18}\)
\(x = \cfrac{-14 \pm \sqrt{988}}{-18}\)
Using \(\pm\), we can determine the roots either by adding or subtracting the \(2\) values in the numerator.
We can determine the first root by adding the values in the numerator:
\(x_1 = \cfrac{-14 + \sqrt{988}}{-18}\)
\(x_1 = 2.52\)
We can determine the second root by subtracting the values in the numerator:
\(x_2 = \cfrac{-14 - \sqrt{988}}{-18}\)
\(x_2 = -0.97\)
Therefore, we can determine that the tangent slope is \(0\) at \(\boldsymbol{x = -0.97, 2.52}\). These points also indicate where the turning points are.
vi. We can determine the \(y\)-coordinates of the turning points by substituting the corresponding roots into the original equation:
\(y_1 = -3(2.52)^3 + 7(2.52)^2 + 22(2.52) -8\)
\(y_1 = 43.88\)
\(y_2 = -3(-0.97)^3 + 7(-0.97)^2 + 22(-0.97) -8\)
\(y_2 = -20.015\)
Therefore, we can determine that the turning points are located at approximately \(\boldsymbol{(2.52, 43.88)}\) and \(\boldsymbol{(-0.97, -20.02)}\).
vii. Given all the information we have determined about the function, we can sketch a graph of the function as such:
