As noted in the introduction to this chapter, the graphs of polynomial functions have a variety of different shapes, so they are useful for modeling curves. In addition, polynomials are easy to evaluate, so they can be used to compute approximate values for other functions, such as exponentials and logarithms. In this section we’ll learn some of the algebraic techniques for studying and using polynomials.
Each of the polynomials above is written in descending powers, which means that the highest-degree term comes first, and the degrees of the terms decrease from largest to smallest. Sometimes it is useful to write a polynomial in ascending powers, so that the degrees of the terms increase. For example, the polynomial \(f(x)\) above would be written as
are \(4\) and \(-7\text{.}\) But now notice that the product of the two linear polynomials \(x-4\) and \(x+7\) is a quadratic polynomial. Or, in other words, the product of two first-degree polynomials gives us a second-degree polynomial. In fact, whenever we multiply two or more polynomials together, we get another polynomial of higher degree.
In part (a) of the Example above, we multiplied a polynomial of degree one by a polynomial of degree three, and the product was a polynomial of degree four. In part (b) of the Example, the product of three first-degree polynomials is a third-degree polynomial.
The degree of \(P\) is 4, and the degree of \(Q\) is 3, so the degree of their product is \(4 + 3 = 7\text{.}\) The only degree 7 term of the product is \((5x^4)(3x^3) = 15x^7\text{,}\) which has coefficient 15.
In the product, each term of \(P(x)\) is multiplied by each term of \(Q(x)\text{.}\) How might we isolate the products that result in terms sof degree 3? We get degree 3 terms by multiplying together terms of degree 0 and 3, or of degree 1 and 2. For these polynomials, the possible combinations are:
The volume of the box in Example 9, \(V = x^3-6x\text{,}\) is an example of a cubic polynomial , a polynomial of degree 3. Cubic polynomials are often used in economics to model cost functions. The cost of producing \(x\) items is an increasing function of \(x\text{,}\) but its rate of increase is usually not constant.
Fixed costs are given by \(C(0) = 250\text{,}\) or $250. The fixed costs include expenses like utility bills that must be paid even if no magazines are produced.
The graph is shown in figure (a). It is increasing from a vertical intercept of 250. The graph is concave down for \(x \lt 8\) approximately, and concave up for \(x\gt 8\text{.}\)
We will solve the equation graphically, as shown in figure (b). Graph \(y = 1200\) along with the cost function, and use the intersect command to find the intersection point of the graphs, \((15.319, 1200)\text{.}\)\(C(x) = 1200\) when \(x\) is about 15,319, so 15,319 copies can be printed for $1200.
Although the cost is always increasing, it increases very slowly from about \(x = 5\) to about \(x = 11\text{.}\) The flattening of the graph in this interval is a result of economy of scale: By buying supplies in bulk and using time efficiently, the cost per magazine can be minimized. However, if the production level is too large, costs begin to rise rapidly again.
In Example 8.1.11c, we solved a cubic equation graphically. There is a cubic formula, analogous to the quadratic formula, that allows us to solve cubic equations algebraically, but it is complicated and not often used in applications.
Leon is flying his plane to Au Gres, Michigan. He maintains a constant altitude until he passes over a marker just outside the neighboring town of Omer, when he begins his descent for landing. During the descent, his altitude, in feet, is given by
Did you ever wonder how your calculator obtains decimal values for numbers such as \(\sqrt{1.1}~\) or \(10^{0.8}~\text{?}\) One way to do it is to use a polynomial. Functions such as \(f(x) = \sqrt{x}~\) and \(f(x) = 10^x~\) can be approximated by polynomials, and the higher the degree of the polynomial, the better the approximation. Because evaluating a polynomial uses only addition and multiplication, calculators and computers can perform the calculations quickly and give us decimal values with great accuracy.
A calculator returns the value \(\sqrt{1.1} = 1.048808848\text{.}\) Our approximation is accurate to four decimal places. One can use a polynomial of higher degree to obtain an approximation with greater accuracy.
As we saw in Chapter 3, many quadratic polynomials can be factored into a product of two linear expressions, and these factors give us information about the roots of a quadratic equation or the \(x\)-intercepts of its graph. The same is true of cubic polynomials, so we will take a brief look at how they can be factored.
Note the difference between the expressions \(~(x+y)^3~\) and \(~x^3 + y^3~\text{;}\) they are not the same! You can see this easily if you refer back to the box for the cube of a binomial. You already know that \(~(x+y)^2~\) is not equivalent to \(~x^2 + y^2~\) because \(~(x+y)^2 = x^2 + 2xy + y^2~\text{,}\) and in fact for any power \(n\) (except 1) the expressions \(~(x+y)^n~\) and \(~x^n + y^n~\) are not equivalent.
This polynomial is a sum of two cubes. The cubed expressions are \(2a\text{,}\) because \((2a)^3 = 8a^3\text{,}\) and \(b\text{.}\) Use formula 1 as a pattern, replacing \(x\) with \(\alert{2a}\text{,}\) and \(y\) with \(\blert{b}\text{.}\)
This polynomial is a difference of two cubes. The cubed expressions are \(1\text{,}\) because \(1^3 = 1\text{,}\) and \(3h^2\text{,}\) because \((3h^2)^3 = 27h^6\text{.}\) Use formula 2 above as a pattern, replacing \(x\) by \(\alert{1}\text{,}\) and \(y\) by \(\blert{3h^2}\text{:}\)
A Bézier curve is actually a sequence of short curves pieced together. Each piece has two endpoints and (for nonlinear curves) at least one control point. The control points do not lie on the curve itself, but they determine its shape. Two polynomials define the curve, one for the \(x\)-coordinate and one for the \(y\)-coordinate.
The linear Bézier curve for two endpoints, \((x_1, y_1)\) and \((x_2, y_2)\text{,}\) is the straight line segment joining those two points. The curve is defined by the two functions
\begin{align*}
x \amp= f(t) = x_1\cdot (1 - t) + x_2\cdot t \\
y \amp = g(t) = y_1 \cdot (1 - t) + y_2 \cdot t
\end{align*}
Find the functions \(f\) and \(g\) defining the linear Bézier curve joining the two points \((-4,7)\) and \((20)\text{.}\) Simplify the formulas defining each function.
Find the functions \(f\) and \(g\) for the quadratic Bézier curve defined by the endpoints \((-4, 7)\) and \((2, 0)\text{,}\) and the control point \((0, 5)\text{.}\) Simplify the formulas defining each function.
We can adjust the curvature of the diagonal stroke of the 7 by moving the control point. Find the functions \(f\) and \(g\) for the quadratic Bézier curve defined by the endpoints \((4,7)\) and \((0,-7)\text{,}\) and the control point \((0,-3)\text{.}\) Simplify the formulas defining each function.
On your graphs in steps (5) and (8), plot the three points that defined the curved section of the numeral 7, then connect them in order with line segments. How does the position of the control point change the curve?
A cubic Bézier curve is defined by two endpoints, \((x_1,y_1)\) and \((x_4,y_4)\text{,}\) and two control points, \((x_2,y_2)\) and \((x_3,y_3)\text{.}\)
Find the functions \(f\) and \(g\) for the cubic Bézier curve defined by the endpoints \((4,7)\) and \((-4,-5)\text{,}\) and the control points \((3,3)\) and \((0,-8)\text{.}\) Simplify the formulas defining each function.
Connect the four given points in order using three line segments. How does the position of the control points affect the curve? Finish the letter y by including the linear Bézier curve you drew for step (2).
A paper company plans to make boxes without tops from sheets of cardboard 12 inches wide and 16 inches long. The company will cut out four squares of side \(x\) inches from the corners of the sheet and fold up the edges as shown in the figure.
A grain silo is built in the shape of a cylinder with a hemisphere on top (see the figure). Write an expression for the volume of the silo in terms of the radius and height of the cylindrical portion of the silo.
A doctor who is treating a heart patient wants to prescribe medication to lower the patient’s blood pressure. The body’s reaction to this medication is a function of the dose administered. If the patient takes \(x\) milliliters of the medication, his blood pressure should decrease by \(R = f (x)\) points, where
\begin{equation*}
f (x) = 3x^2 - \dfrac{1}{3}x^3
\end{equation*}
There may be risks associated with a large change in blood pressure. How many milliliters of the medication should be administered to produce half the maximum possible drop in blood pressure?
where \(t\) is the number of hours since the high tide. The approximation is valid for \(-4 \le t \le 4\text{.}\) (A negative value of \(t\) corresponds to a number of hours before the high tide.)
Graph the polynomial for \(-4 \le t \le 4\text{.}\)
During an earthquake, Nordhoff Street split in two, and one section shifted up several centimeters. Engineers created a ramp from the lower section to the upper section. In the coordinate system shown in the figure below, the ramp is part of the graph of
\begin{equation*}
y = f (x) = -0.00004x^3 - 0.006x^2 + 20
\end{equation*}
