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Find a degree 
Magazine Chapter 4: Problem 59
\(59-62\) . Find a polynomial of the specified degree that has the given zeros. Degree \(3 ; \quad\) zeros \(-1,1,3\)
Short Answer
Expert verified
The polynomial is \(x^3 - 3x^2 - x + 3\).
Step by step solution
01
Understand Polynomial Degree
We need to construct a polynomial of degree 3. A polynomial of degree 3 can be written as a product of three linear factors.
02
Set up Factors from Zeros
The zeros of the polynomial are given as \(-1, 1,\) and \(3\). Hence, the polynomial can be written as the product of linear binomials: \[(x + 1)(x - 1)(x - 3)\]
03
Expand the First Two Factors
Multiply the first two factors:\[(x + 1)(x - 1) = x^2 - 1\]This is a difference of squares.
04
Multiply by the Remaining Factor
Now multiply the result by the third factor:\[ (x^2 - 1)(x - 3) = x^2(x) - x^2(3) - 1(x) + 1(3) \] \[ = x^3 - 3x^2 - x + 3 \]
05
Write the Final Polynomial
Thus, the polynomial of degree 3 with zeros \(-1, 1,\) and \(3\) is \[x^3 - 3x^2 - x + 3\]
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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 key feature that helps us understand its shape and complexity. In simple terms, the degree of a polynomial is the highest power of the variable in the expression. For instance, in the polynomial \(x^3 - 3x^2 - x + 3\), the degree is 3 because the highest power of \(x\) is 3.
The degree gives us important clues about the polynomial's behavior:
The degree gives us important clues about the polynomial's behavior:
- A polynomial of degree 1 is a straight line.
- A polynomial of degree 2 forms a parabola.
- A polynomial of degree 3 creates a cubic curve, which can twist and turn.
Zeros of a Polynomial
Zeros of a polynomial, also known as roots or solutions, are the values of \(x\) that make the polynomial equal to zero. If you substitute a zero into the polynomial, the result will be zero. These are crucial because they are the points where the graph of the polynomial will intersect the x-axis.
For the exercise, the zeros provided are \(-1\), \(1\), and \(3\). This means our polynomial will hit the x-axis at these x-values. To construct such a polynomial, we start by forming linear factors:
For the exercise, the zeros provided are \(-1\), \(1\), and \(3\). This means our polynomial will hit the x-axis at these x-values. To construct such a polynomial, we start by forming linear factors:
- For zero \(-1\), the factor is \((x + 1)\).
- For zero \(1\), the factor is \((x - 1)\).
- For zero \(3\), the factor is \((x - 3)\).
Expanding Polynomial Expressions
Expanding polynomial expressions involves simplifying a product of polynomials into a standard polynomial form. This process requires distributing and combining like terms to get the final expression. In this case, combining factors like \((x + 1)(x - 1)(x - 3)\) is necessary to form a comprehensive polynomial.
The task given first uses the identity of the difference of squares to expand \((x + 1)(x - 1)\) into \(x^2 - 1\). Then, this result is multiplied by the remaining factor \((x - 3)\).
The expansion follows:
The task given first uses the identity of the difference of squares to expand \((x + 1)(x - 1)\) into \(x^2 - 1\). Then, this result is multiplied by the remaining factor \((x - 3)\).
The expansion follows:
- Multiply \(x^2\) by each term in \((x - 3)\), giving \(x^3 - 3x^2\).
- Multiply \(-1\) by each term in \((x - 3)\), resulting in \(-x + 3\).
Factorization of Polynomials
Factorization is breaking down a complex polynomial into simpler, multiplied parts called factors. These factors return the original polynomial when multiplied together. Factorization is highly useful when you want to find the zeros of a polynomial or simplify equations.
In this exercise, we formulated our polynomial by initially considering its zeros. Starting from the zeros, we developed factors like \((x + 1)(x - 1)(x - 3)\), which correspond to each root.
This process:
In this exercise, we formulated our polynomial by initially considering its zeros. Starting from the zeros, we developed factors like \((x + 1)(x - 1)(x - 3)\), which correspond to each root.
This process:
- Helps in quickly identifying the zeros of a polynomial.
- Allows us to reconfirm the degree鈥攕ince each factor corresponds to a root.
- Makes operations like division by polynomials and finding remainders simpler.
