Algebra Worksheet Section 10.6 Factoring Polynomials Of The Form Answers
Vaniria Setser
where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.
Algebra Worksheet Section 10.6 Factoring Polynomials Of The Form Answers
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The x-intercept [latex]x=-3[/latex] is the solution to the equation [latex]\left(x+3\right)=0[/latex]. The graph passes directly through the x-intercept at [latex]x=-3[/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The factor is repeated, that is, the factor [latex]\left(x - 2\right)[/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice.
The x-intercept [latex]x=-1[/latex] is the repeated solution of factor [latex]\left(x+1\right)^3=0[/latex]. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function [latex]f\left(x\right)=x^3[/latex]. We call this a triple zero, or a zero with multiplicity 3.
For zeros with even multiplicities, the graphs touch or are tangent to the x-axis at these x-values. For zeros with odd multiplicities, the graphs cross or intersect the x-axis at these x-values. See the graphs below for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.
For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the x-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the x-axis.
If a polynomial contains a factor of the form [latex]\left(x-h\right)^p[/latex], the behavior near the x-intercept h is determined by the power p. We say that [latex]x=h[/latex] is a zero of multiplicity p.
The last zero occurs at [latex]x=4[/latex]. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is 3 and that the sum of the multiplicities must be 6.
Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]a_nx^n[/latex], is an even power function, as x increases or decreases without bound, [latex]f\left(x\right)[/latex] increases without bound. When the leading term is an odd power function, as x decreases without bound, [latex]f\left(x\right)[/latex] also decreases without bound; as x increases without bound, [latex]f\left(x\right)[/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below summarizes all four cases.
We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.
Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2x^3[/latex], so the end behavior, as seen in the following graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.
In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. Consider a polynomial function f whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers a and b in the domain of f, if a < b and [latex]f\left(a\right)\ne f\left(b\right)[/latex], then the function f takes on every value between [latex]f\left(a\right)[/latex] and [latex]f\left(b\right)[/latex].
We can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function f at [latex]x=a[/latex] lies above the x-axis and another point at [latex]x=b[/latex] lies below the x-axis, there must exist a third point between [latex]x=a[/latex] and [latex]x=b[/latex] where the graph crosses the x-axis. Call this point [latex]\left(c,\text f\left(c\right)\right)[/latex]. This means that we are assured there is a value c where [latex]f\left(c\right)=0[/latex].
In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis. The figure below shows that there is a zero between a and b.
Let f be a polynomial function. The Intermediate Value Theorem states that if [latex]f\left(a\right)[/latex] and [latex]f\left(b\right)[/latex] have opposite signs, then there exists at least one value c between a and b for which [latex]f\left(c\right)=0[/latex].
We see that one zero occurs at [latex]x=2[/latex]. Also, since [latex]f\left(3\right)[/latex] is negative and [latex]f\left(4\right)[/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.
Because f is a polynomial function and since [latex]f\left(1\right)[/latex] is negative and [latex]f\left(2\right)[/latex] is positive, there is at least one real zero between [latex]x=1[/latex] and [latex]x=2[/latex].Writing Formulas for Polynomial FunctionsNow that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors.
If a polynomial of lowest degree p has zeros at [latex]x=x_1,x_2,\dots ,x_n[/latex], then the polynomial can be written in the factored form: [latex]f\left(x\right)=a\left(x-x_1\right)^p_1\left(x-x_2\right)^p_2\cdots \left(x-x_n\right)^p_n[/latex] where the powers [latex]p_i[/latex] on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the x-intercept.
With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. These are also referred to as the absolute maximum and absolute minimum values of the function.
A local maximum or local minimum at x = a (sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around x = a. If a function has a local maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x in an open interval around x = a. If a function has a local minimum at a, then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x in an open interval around x = a.
A global maximum or global minimum is the output at the highest or lowest point of the function. If a function has a global maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x. If a function has a global minimum at a, then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x.
An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.
Notice that after a square is cut out from each end, it leaves a [latex]\left(14 - 2w\right)[/latex] cm by [latex]\left(20 - 2w\right)[/latex] cm rectangle for the base of the box, and the box will be w cm tall. This gives the volume
Notice, since the factors are w, [latex]20 - 2w[/latex] and [latex]14 - 2w[/latex], the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values w may take on are greater than zero or less than 7. This means we will restrict the domain of this function to [latex]0
On this graph, we turn our focus to only the portion on the reasonable domain, [latex]\left[0,\text 7\right][/latex]. We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph below.
In previous lessons, students have learned about how the factored form of a polynomial shows the x-intercepts of the polynomial. In the last lesson, they were reminded that a factor of a number means that the factor divides the number evenly. For example, 6 is a factor of 24 because 246=424\div 6=4246=4. Additionally, we know from this division that 4 is also a factor. We're going to use these concepts today to guide students to an understanding that if a polynomial divides another evenly then it is a factor. So we can use polynomial division to find the factors of a polynomial which helps us to factor a polynomial or to write it in factored form. Also, as we are finding the factors, we are finding the x-intercepts of the polynomial. In the next lesson, we will use all of this to focus on solving polynomials instead of factoring them.
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