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Chapter 3: Problem 77

Write a polynomial \(f(x)\) that meets the given conditions. Answers may vary. (See Example 10 ) Degree 3 polynomial with zeros \(-2,3 i,\) and \(-3 i\).

Short Answer

Expert verified
The polynomial is f(x) = x^3 + 2x^2 - 9x - 18.

Step by step solution

01

Identify the zeros

The given zeros are -2, 3i, and -3i.
02

Write the factors

For each zero, write its corresponding factor: If -2 is a zero, the factor is (x + 2). If 3i is a zero, the factor is (x - 3i). If -3i is a zero, the factor is (x + 3i).
03

Multiply the complex factors

First, multiply the factors involving i: (x - 3i)(x + 3i). This simplifies using the difference of squares: (x^2 - (3i)^2) = (x^2 - 9).
04

Multiply all factors

Now, multiply (x + 2) by (x^2 - 9): (x + 2)(x^2 - 9). Using distribution (FOIL method): (x)(x^2) + (x)(-9) + (2)(x^2) + (2)(-9) = x^3 - 9x + 2x^2 - 18. Arrange in standard polynomial form: x^3 + 2x^2 - 9x - 18.

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Key Concepts

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

degree of polynomial
The degree of a polynomial is a fundamental concept. It is the highest power of the variable in the polynomial expression. For instance, in the exercise, we have a polynomial with a degree 3, because the highest power of the variable x is 3. This tells us that there are three roots (or zeros) for the polynomial. Polynomials are named based on their degrees:
  • Degree 1: Linear polynomial - e.g., \(ax + b\)
  • Degree 2: Quadratic polynomial - e.g., \(ax^2 + bx + c\)
  • Degree 3: Cubic polynomial - e.g., \(ax^3 + bx^2 + cx + d\)
Understanding the degree helps in factoring and solving polynomial equations.
complex zeros
Complex zeros are solutions to a polynomial equation that have both real and imaginary parts. Here, we have complex zeros \(3i\) and \(-3i\). Complex numbers are in the form \(a + bi\), where \(i\) is the imaginary unit with the property that \(i^2 = -1\). Complex zeros often come in conjugate pairs (e.g., \(3i\) and \(-3i\)), ensuring that when multiplied, the imaginary parts cancel out. This affects how we write and simplify polynomial factors. For example, multiplying \((x - 3i)(x + 3i)\) results in a real polynomial expression \(x^2 + 9\).
standard polynomial form
A polynomial in standard form is arranged by the terms' degrees in descending order. Each term consists of a coefficient and a power of the variable. The polynomial from the solution is \(x^3 + 2x^2 - 9x - 18\), which is already in standard form:
  • \(x^3\) (cubic term)
  • \(2x^2\) (quadratic term)
  • \(-9x\) (linear term)
  • \(-18\) (constant term)
By arranging the terms in descending order, the polynomial becomes easily recognizable and ready for further operations like differentiation or evaluation.
distribution method
The distribution method, often called the FOIL (First, Outer, Inner, Last) method, is a technique to expand binomials. This method ensures that each term in one binomial is multiplied by each term in the other binomial. For example, in the exercise, multiplying \((x + 2)(x^2 - 9)\) involves:
\((x + 2)(x^2 - 9) = x(x^2) + x(-9) + 2(x^2) + 2(-9) = x^3 - 9x + 2x^2 - 18\).
This detailed multiplication helps in systematically expanding and simplifying expressions, leading to the correct polynomial form.
difference of squares
The difference of squares is a handy algebraic identity used to simplify expressions involving squares. It states that \(a^2 - b^2 = (a - b)(a + b)\). In the exercise, we use this identity to simplify the product of complex factors:
\((x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - 9\).
Here, \(a = x\) and \(b = 3i\). The identity helps us remove the imaginary unit, turning the expression into a real polynomial. This method is widely applicable for various problems involving squares.

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Most popular questions from this chapter

The number of U.S. citizens of voting age, \(N(t)\) (in millions), can be modeled according to the number of years \(t\) since 1932 . $$N(t)=0.0135 t^{2}+1.09 t+73.2$$ The function defined by $$V(t)=1.08 t+36.9$$ represents the number of people who voted \(V(t)\) (in millions) in U.S. presidential elections ( \(t\) is the number of years since 1932 and \(t\) is a multiple of 4). (Source: U.S. Census Bureau, www.census.gov) a. Write the function defined by \(P(t)=\left(\frac{V}{N}\right)(t)\) and interpret its meaning in the context of this problem. b. Evaluate \(P(60)\) and interpret its meaning in the context of this problem. Round to 2 decimal places.

a. Write an equation for a rational function \(f\) whose graph is the same as the graph of \(y=\frac{1}{x^{2}}\) shifted up 3 units and to the left 1 unit. b. Write the domain and range of the function in interval notation.

Determine if the statement is true or false. Given \(f(x)=2 i x^{4}-(3+6 i) x^{3}+5 x^{2}+7,\) if \(a+b i\) is a zero of \(f(x)\), then \(a-b i\) must also be a zero.

Determine if the statement is true or false. If 5 is an upper bound for the real zeros of \(f(x)\), then 6 is also an upper bound.

A food company originally sells cereal in boxes with dimensions 10 in. by 7 in. by 2.5 in. To make more profit, the company decreases each dimension of the box by \(x\) inches but keeps the price the same. If the new volume is \(81 \mathrm{in} .{ }^{3}\) by how much was each dimension decreased?
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