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Magazine Chapter 3: Problem 22
Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$1+\sqrt{2}, 1-\sqrt{2}, \text { and } 3$$
Short Answer
Expert verified
The polynomial is \(P(x) = x^3 - 5x^2 + 7x - 3\).
Step by step solution
01
Understanding the Zeros
The polynomial has zeros at \(1+\sqrt{2}, 1-\sqrt{2},\) and \(3\). Since the coefficients of the polynomial must be real numbers, any irrational roots must occur in conjugate pairs. In this case, \(1+\sqrt{2}\) and \(1-\sqrt{2}\) are conjugates, so we can proceed to forming factors from these zeros.
02
Forming the Factors
For each zero \(c\), the corresponding linear factor of the polynomial is \((x-c)\). Thus, we have the factors as \((x-(1+\sqrt{2})), (x-(1-\sqrt{2})),\) and \((x-3)\).
03
Combining Conjugate Factors
Combine the conjugate factors: \[(x-(1+\sqrt{2}))(x-(1-\sqrt{2})) = ((x-1)-\sqrt{2})((x-1)+\sqrt{2})\]. This expression is a difference of squares: \((a-b)(a+b) = a^2 - b^2\).
04
Simplifying the Difference of Squares
Calculate the difference of squares: \[((x-1)-\sqrt{2})((x-1)+\sqrt{2}) = (x-1)^2 - (\sqrt{2})^2 = x^2 - 2x + 1 - 2\]. Simplifying this yields \(x^2 - 2x - 1\).
05
Form the Complete Polynomial
Now multiply the simplified factor with the remaining factor \((x - 3)\): \[P(x) = (x^2 - 2x - 1)(x - 3)\].
06
Expanding the Polynomial
Expand \((x^2 - 2x - 1)(x - 3)\) using distribution:\[\begin{aligned} P(x) &= x^2(x - 3) - 2x(x - 3) - 1(x - 3) \&= x^3 - 3x^2 - 2x^2 + 6x - x + 3 \&= x^3 - 5x^2 + 7x - 3.\end{aligned}\]
07
Conclusion
Thus, the polynomial with the given properties is \(P(x) = x^3 - 5x^2 + 7x - 3\). It satisfies all conditions: it has a leading coefficient of 1, least degree of 3 (equal to the number of distinct zeros), and real coefficients.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Zeros of Polynomials
In the realm of polynomial functions, zeros hold a fundamental place. A zero of a polynomial is simply a value of the variable for which the polynomial evaluates to zero. In our task, we were given the zeros as \(1+\sqrt{2}, 1-\sqrt{2}\), and \(3\). These represent the values where the polynomial touches or crosses the x-axis.
- Each zero corresponds to a factor of the polynomial. If \(c\) is a zero, then \((x-c)\) is a factor.
- The degree of the polynomial is related to the number of zeros it has. Here, we have three zeros, so the minimum degree of the polynomial is three.
- In polynomials with real coefficients, if one root, especially an irrational one involving a square root, is known, its conjugate must also be a root. Hence, \(1+\sqrt{2}\) and \(1-\sqrt{2}\) are used in pairs.
Real Coefficients
Polynomials with real coefficients are crucial in many practical applications. Real coefficients mean each term in the polynomial is a real number, encompassing both rational and irrational numbers. This property influences the nature and appearance of the polynomial's zeros.
- When coefficients are real, any non-real roots must appear in conjugate pairs if present. For real polynomials, complex numbers' conjugates ensure that the final polynomial expression does not include imaginary components.
- An example of this property in action is using \(1+\sqrt{2}\) and \(1-\sqrt{2}\). Although \(\sqrt{2}\) is irrational, its conjugate pairs with its negative counterpart to form a linear combination that results in a real coefficient polynomial.
- This system ensures simpler interpretation and application of the polynomial function in real-life contexts, including physics and engineering.
Difference of Squares
The difference of squares is a fundamental algebraic identity used to simplify expressions involving squared terms subtracted from each other. This identity is expressed as \((a-b)(a+b) = a^2 - b^2\). It serves as a handy tool when dealing with roots, especially quadratic roots like \(\sqrt{2}\) as seen in our solution.
- The difference of squares helps simplify conjugate factor expressions, such as \( (x-(1+\sqrt{2}))(x-(1-\sqrt{2})) \).
- By recognizing this pattern, we simplify to \((x-1)^2 - (\sqrt{2})^2 = x^2 - 2x + 1 - 2\), thus cleanly reducing down to a simpler quadratic form \( x^2 - 2x - 1 \).
- Utilizing this technique greatly reduces computational effort and helps maintain accuracy. It's a strategic approach in polynomial manipulations.
