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Find a degree 
Magazine Chapter 3: Problem 61
Find a polynomial of degree 3 that has zeros \(1,-2,\) and 3 and in which the coefficient of \(x^{2}\) is 3.
Short Answer
Expert verified
\( f(x) = -\frac{3}{2}x^3 + 3x^2 + \frac{15}{2}x - 9 \)
Step by step solution
01
Form the Basic Polynomial
A polynomial of degree 3 can be expressed as \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \]where \( r_1, r_2, \) and \( r_3 \) are the roots of the polynomial. Given the roots are 1, -2, and 3, the polynomial can be initially written as:\[ f(x) = a(x - 1)(x + 2)(x - 3) \]
02
Expand the Factors
Expand the factors \((x - 1)(x + 2)(x - 3)\) to express the polynomial in standard form. Start by multiplying the first two factors:\[ (x - 1)(x + 2) = x(x + 2) - 1(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2 \]Next, multiply the result by the remaining factor \((x - 3)\):\[ (x^2 + x - 2)(x - 3) = x^3 - 3x^2 + x^2 - 3x - 2x + 6 = x^3 - 2x^2 - 5x + 6 \]
03
Apply the Coefficient Condition
We know the coefficient of \(x^2\) should be 3. The polynomial from Step 2 is:\[ f(x) = a(x^3 - 2x^2 - 5x + 6) \]To make the coefficient of \(x^2\) equal to 3, solve for \(a\):\[ -2a = 3 \]\[ a = -\frac{3}{2} \]
04
Write the Final Polynomial
Substitute \(a = -\frac{3}{2}\) into the expanded polynomial to obtain the final polynomial:\[ f(x) = -\frac{3}{2}(x^3 - 2x^2 - 5x + 6) \]Distribute \(-\frac{3}{2}\) through the terms:\[ f(x) = -\frac{3}{2}x^3 + 3x^2 + \frac{15}{2}x - 9 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Zeros of a Polynomial
Identifying the zeros of a polynomial is an essential step in determining its overall behavior and properties. Zeros of a polynomial, also known as roots, are the values of the variable that make the polynomial equal to zero.
For a cubic polynomial, meaning a polynomial of degree 3, there are generally three zeros. Given the zeros like in this exercise:
For a cubic polynomial, meaning a polynomial of degree 3, there are generally three zeros. Given the zeros like in this exercise:
- 1
- -2
- 3
Coefficient Matching
Coefficient matching involves adjusting the constant factor in a polynomial so that it fits a given condition, often related to the coefficients of certain terms. In the context of this exercise, we focus on the coefficient of \( x^2 \).
We develop an expression for the polynomial through expansion and look at the \( x^2 \) term. This polynomial is initially \( f(x) = a(x^3 - 2x^2 - 5x + 6) \). For the coefficient matching condition, we set the coefficient of \( x^2 \) to 3.
Applying the Condition:
We develop an expression for the polynomial through expansion and look at the \( x^2 \) term. This polynomial is initially \( f(x) = a(x^3 - 2x^2 - 5x + 6) \). For the coefficient matching condition, we set the coefficient of \( x^2 \) to 3.
Applying the Condition:
- Identify the existing \( x^2 \) coefficient as \(-2a\).
- Align this with the desired coefficient: \(-2a = 3\).
- Solve for \( a \) to find \( a = -\frac{3}{2} \).
Polynomial Expansion
Polynomial expansion is the process where you multiply terms together to express a polynomial in its standard form. This is crucial when converting a polynomial given by its factors into an understandable form with explicit coefficients for each degree.
In this case, multiply \((x - 1)(x + 2)(x - 3)\):
In this case, multiply \((x - 1)(x + 2)(x - 3)\):
- First, multiply \((x - 1)\) and \((x + 2)\), obtaining \(x^2 + x - 2\).
- Then multiply \((x^2 + x - 2)\) by \((x - 3)\) to get the fully expanded form of \(x^3 - 2x^2 - 5x + 6\).
Cubic Polynomial
A cubic polynomial is a type of polynomial with a degree of 3, indicating its highest term is raised to the power of 3. These polynomials form a family that can model a wide variety of behaviors, such as polynomial curves with distinct peaks and troughs.
In this exercise, we derive a cubic polynomial with specified properties. Starting from the zeros 1, -2, and 3, we follow a mechanical process of factor multiplication and coefficient adjustment to ensure it meets the required form.
Key Characteristics:
In this exercise, we derive a cubic polynomial with specified properties. Starting from the zeros 1, -2, and 3, we follow a mechanical process of factor multiplication and coefficient adjustment to ensure it meets the required form.
Key Characteristics:
- Three roots, representing intersections with the x-axis, previously identified as 1, -2, and 3.
- The standard form after factoring and expanding: \(x^3 - 2x^2 - 5x + 6\).
- Adjustment as \(-\frac{3}{2}x^3 + 3x^2 + \frac{15}{2}x - 9\) to match the given conditions on coefficients.
