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Chapter 3: Problem 60

Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros -2,0,2,4

Short Answer

Expert verified
The polynomial is \(P(x) = x^4 - 4x^3 - 4x^2 + 16x\).

Step by step solution

01

Understand the Zeros

The given zeros of the polynomial are -2, 0, 2, and 4. These values are the roots of the polynomial equation. Since we are finding a polynomial of degree 4, each zero corresponds to a factor of the polynomial.
02

Write the Polynomial in Factored Form

Using the zeros, write the polynomial in its factored form. Each zero \(-2, 0, 2, 4 \) corresponds to a factor: \(x+2, x, x-2, \) and \(x-4\). Thus, the polynomial can be expressed as:\[P(x) = (x + 2)(x)(x - 2)(x - 4)\]
03

Multiply the Factors

Start multiplying the factors to find the polynomial in expanded form. Begin by multiplying the first two factors, followed by the last two, and then the two resulting expressions:\[(x+2)(x) = x^2 + 2x\]\[(x-2)(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8\]
04

Expand the Polynomial

Multiply the two quadratic expressions obtained in the previous step:\[(x^2 + 2x)(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 + 2x^3 - 12x^2 + 16x\]Combine like terms to simplify:\[x^4 - 4x^3 - 4x^2 + 16x\]
05

Write the Final Polynomial

The fully expanded polynomial is:\[P(x) = x^4 - 4x^3 - 4x^2 + 16x\] This is the polynomial of degree 4 with the given zeros.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factored Form
When a polynomial is expressed in a way that illustrates its roots (or zeros), it is said to be in its "factored form." This representation is beneficial because it shows exactly where the graph of the polynomial will intersect the x-axis.
  • In the example given, the zeros are -2, 0, 2, and 4. Each of these zeros gives us a linear factor:
    • -2 corresponds to the factor \(x+2\)
    • 0 leads to the factor \(x\)
    • 2 gives the factor \(x-2\)
    • 4 results in the factor \(x-4\)
Formally speaking, if \( r \) is a root of the polynomial, then \((x - r)\) is a factor. Therefore, a polynomial of degree 4 with these zeros can be expressed as:\[ P(x) = (x+2)(x)(x-2)(x-4) \]This factored form is particularly insightful when considering the behavior of polynomials, as it makes influences such as multiplicity of roots clear.
Polynomial Expansion
Polynomial expansion involves taking a polynomial in its factored form and expanding it into a sum of terms. This process of expanding allows us to clearly see the degree and the leading coefficients which tell us a lot about the shape and orientation of the polynomial's graph.
  • In the example solution, expansion was started by multiplying pairs of linear factors to convert the polynomial into quadratic expressions.
    • First, \( (x+2)(x) = x^2 + 2x \)
    • Then, \( (x-2)(x-4) = x^2 - 6x + 8 \)
The final step is to multiply these quadratic expressions to achieve a single polynomial:\[ (x^2 + 2x)(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 + 2x^3 - 12x^2 + 16x \]Combining like terms leads to:\[ x^4 - 4x^3 - 4x^2 + 16x \]This is the polynomial in its expanded form, making it easier to analyze for calculus purposes such as derivative and integral calculations.
Zeros of Polynomial
The zeros of a polynomial are the values at which the polynomial evaluates to zero. In simpler terms, they are the x-values where the graph of the polynomial touches or crosses the x-axis.
  • For the polynomial in question, the zeros given were -2, 0, 2, and 4.
    • These zeros translated directly into the factors \(x+2\), \(x\), \(x-2\), and \(x-4\).
    • If a graph of \(P(x)\) were drawn, it would touch or cross the x-axis at these points.
Zeros are fundamental in both algebra and calculus as they provide solutions to the polynomial equation \(P(x) = 0\). Understanding zeros offers insights into the polynomial's shape and structure, its potential maxima and minima, and even helps in integration calculations.

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Most popular questions from this chapter

The quadratic formula can be used to solve any quadratic (or second-degree) equation. You might have wondered whether similar formulas exist for cubic (thirddegree), quartic (fourth-degree), and higher-degree equations. For the depressed cubic \(x^{3}+p x+q=0,\) Cardano (page 274) found the following formula for one solution: $$x=\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{\frac{-q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}$$ A formula for quartic equations was discovered by the Italian mathematician Ferrari in \(1540 .\) In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossible to write a quintic formula, that is, a formula for fifth-degree equations. Finally, Galois (page 254) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the cubic formula to find a solution for the following equations. Then solve the equations using the methods you learned in this section. Which method is easier? (a) \(x^{3}-3 x+2=0\) (b) \(x^{3}-27 x-54=0\) (c) \(x^{3}+3 x+4=0\)

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}-3 x+3=0$$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$x^{4}-x-4=0$$

Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ h(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

Suppose that the rabbit population on Mr. Jenkins' farm follows the formula $$ p(t)=\frac{3000 t}{t+1} $$ where \(t \geq 0\) is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population. (b) What eventually happens to the rabbit population? (IMAGES CANNOT COPY)
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