Problem 2
Question
Find a cubic polynomial in standard form with real coefficients, having the given zeros. Let the leading coefficient be \(1 .\) Do not use a calculator. \(-3\) and \(6+2 i\)
Step-by-Step Solution
Verified Answer
The cubic polynomial is \(x^3 - 9x^2 + 4x + 120\).
1Step 1: Identify Given Zeros
We are provided with two zeros, \(-3\) and \(6+2i\). Since we want a cubic polynomial and the coefficients need to be real, the conjugate of the complex root must also be a zero. Thus, the zeros are \(-3\), \(6+2i\), and \(6-2i\).
2Step 2: Write the Polynomial Factors
Based on the identified zeros, the polynomial can be expressed as a product of factors: \((x + 3)(x - (6+2i))(x - (6-2i))\).
3Step 3: Multiply Complex Conjugate Factors
Compute the product of the complex conjugate factors: \((x - (6+2i))(x - (6-2i)) = ((x - 6) - 2i)(x - 6 + 2i)\).This results in a difference of squares: \((x-6)^2 - (2i)^2\).
4Step 4: Simplify the Complex Product
Simplify the expression from the previous step:\((x-6)^2 - (-4)\)\[(x-6)^2 + 4\]\[(x^2 - 12x + 36) + 4 \]\[x^2 - 12x + 40\].
5Step 5: Multiply All Factors
Multiply the result from Step 4 by \(x + 3\):\((x + 3)(x^2 - 12x + 40)\).
6Step 6: Expand the Polynomials
Expand the expression:First, distribute \(x\):\[x(x^2 - 12x + 40) = x^3 - 12x^2 + 40x\]Next, distribute \(3\):\[3(x^2 - 12x + 40) = 3x^2 - 36x + 120\]Combine the results:\[x^3 - 12x^2 + 40x + 3x^2 - 36x + 120\]Combine like terms:\[x^3 - 9x^2 + 4x + 120\].
7Step 7: Write Final Polynomial
The final cubic polynomial in standard form (with leading coefficient 1) is: \[x^3 - 9x^2 + 4x + 120\].
Key Concepts
Complex ConjugatesPolynomial ZerosStandard Form of Polynomial
Complex Conjugates
Complex conjugates are an important concept when dealing with complex numbers, especially in polynomials with real coefficients. A complex number typically has a real part and an imaginary part, such as in the format of \( a + bi \), where \( a \) and \( b \) are real numbers.
The complex conjugate of this number is \( a - bi \). The conjugate simply flips the sign of the imaginary component. For example, if you have a zero \( 6 + 2i \), its complex conjugate is \( 6 - 2i \).
Including the complex conjugate as a zero is crucial when the polynomial has real coefficients. This helps to ensure that the imaginary parts cancel out during multiplication, leaving a product that is a sum of squares, ultimately yielding a real number.
This requirement of conjugates is crucial because if the polynomial is to maintain real coefficients, every complex root must be paired with its conjugate.
The complex conjugate of this number is \( a - bi \). The conjugate simply flips the sign of the imaginary component. For example, if you have a zero \( 6 + 2i \), its complex conjugate is \( 6 - 2i \).
Including the complex conjugate as a zero is crucial when the polynomial has real coefficients. This helps to ensure that the imaginary parts cancel out during multiplication, leaving a product that is a sum of squares, ultimately yielding a real number.
This requirement of conjugates is crucial because if the polynomial is to maintain real coefficients, every complex root must be paired with its conjugate.
Polynomial Zeros
Zeros of a polynomial are the values of \( x \) that make the polynomial evaluate to zero. These zeros are also referred to as the roots of the polynomial. For a cubic polynomial, you will usually need three zeros.
In the polynomial formation process, the given zeros (-3, 6+2i) and the conjugate of the complex root (6-2i) were used. These zeros tell us the \( x \)-intercepts when the polynomial is plotted on a graph.
From these zeros, factors of the form \( (x - ext{zero}) \) are generated. Therefore, the polynomial can be written as the product of these factors:
In the polynomial formation process, the given zeros (-3, 6+2i) and the conjugate of the complex root (6-2i) were used. These zeros tell us the \( x \)-intercepts when the polynomial is plotted on a graph.
From these zeros, factors of the form \( (x - ext{zero}) \) are generated. Therefore, the polynomial can be written as the product of these factors:
- For zero \(-3\), a factor is \((x + 3)\)
- For zero \(6+2i\), a factor is \((x - (6+2i))\)
- For zero \(6-2i\), a factor is \((x - (6-2i))\)
Standard Form of Polynomial
The standard form of a polynomial consists of arranging terms from the highest degree to the lowest degree. For a cubic polynomial, this involves an \( x^3 \) term followed by \( x^2 \), \( x \), and finally a constant.
In the given exercise, after identifying zeros and forming factors, multiplication and expansion were used to convert these into a polynomial in standard form. Initially, complex conjugate factors were multiplied to eliminate the imaginary parts. The expression \((x - (6+2i))(x - (6-2i))\) simplified to \((x^2 - 12x + 40)\).
To achieve the final product, the resulting quadratic expression was multiplied by the remaining linear factor \((x + 3)\). This procedure was done through distributive property, multiplying each term separately, and finally combining like terms for a clean, comprehensive polynomial:
In the given exercise, after identifying zeros and forming factors, multiplication and expansion were used to convert these into a polynomial in standard form. Initially, complex conjugate factors were multiplied to eliminate the imaginary parts. The expression \((x - (6+2i))(x - (6-2i))\) simplified to \((x^2 - 12x + 40)\).
To achieve the final product, the resulting quadratic expression was multiplied by the remaining linear factor \((x + 3)\). This procedure was done through distributive property, multiplying each term separately, and finally combining like terms for a clean, comprehensive polynomial:
- Leading term: \(x^3\)
- Second-degree term: \(-9x^2\)
- First-degree term: \(4x\)
- Constant: \(120\)
Other exercises in this chapter
Problem 2
For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginar
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Find all real solutions. $$x^{4}-x^{3}-6 x^{2}=0$$
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Each expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{6 x^{4}}{2 x^{3}}$$
View solution Problem 2
Match the equation in Column I with its solution(s) in Column II. Do not use a calculator. (II) A. \(\pm 2 i\), B. \(\pm 2 \sqrt{2}\), C. \(\pm i \sqrt{2}\), D.
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