Problem 39
Question
Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(2 ; x=2\) and \(x=-1\) are zeros
Step-by-Step Solution
Verified Answer
The polynomial satisfying the conditions is \(p(x) = x^2-x-2\).
1Step 1: Identify the zeros
We are given two zeros \(x = 2\) and \(x = -1\). These values are the roots of the polynomial, which means when these values are substituted into the polynomial, the result equals 0.
2Step 2: Construct the polynomial
Since the zeros are \(x = 2\) and \(x = -1\), we can use the form \(p(x) = k*(x-a)*(x-b)\) to construct the polynomial. Replace \(a\) with \(2\) and \(b\) with \(-1\) which gives \(p(x) = k*(x-2)*(x+1)\).
3Step 3: Choose the constant
The problem doesn't specify a particular value for \(k\), there may be more than one possible answer. So, we can choose any non-zero constant for \(k\). For simplicity, let's choose \(k = 1\). With this choice, our polynomial becomes \(p(x) = (x-2)*(x+1)\).
4Step 4: Simplify the expression
The final step is to simplify the polynomial expression by multiplying out the brackets, giving \(p(x) = x^2-x-2\).
Key Concepts
Zeros of a PolynomialReal CoefficientsQuadratic Polynomial
Zeros of a Polynomial
Understanding the zeros of a polynomial is fundamental in algebra. Zeros, also known as roots, are the values of \(x\) that make the polynomial equal to zero. These are the points where the graph of the polynomial intersects the x-axis. For example, if you have a polynomial equation \(p(x)\), and when you substitute \(x = 2\) or \(x = -1\) into this equation the result is zero, then 2 and -1 are zeros of \(p(x)\).
This is important because once we know the zeros, constructing the polynomial becomes straightforward. Let's say a polynomial has zeros \(x = a\) and \(x = b\), it can be expressed in the form \((x-a)(x-b)\). This means that by knowing the zeros of the polynomial, you can reverse-engineer the polynomial itself, which is helpful in creating or factoring polynomial expressions.
This is important because once we know the zeros, constructing the polynomial becomes straightforward. Let's say a polynomial has zeros \(x = a\) and \(x = b\), it can be expressed in the form \((x-a)(x-b)\). This means that by knowing the zeros of the polynomial, you can reverse-engineer the polynomial itself, which is helpful in creating or factoring polynomial expressions.
Real Coefficients
Real coefficients refer to the terms in a polynomial that are real numbers. Real numbers include all the numbers on the number line, such as integers, fractions, and irrational numbers. Polynomials with real coefficients mean that every term's coefficient (those numbers in front of \(x\), \(x^2\), etc.) is a real number.
This is a crucial concept in polynomial equations and their factorizations. For example, when a polynomial is expressed as \(p(x) = (x-2)(x+1)\), the coefficients of \(x^2-x-2\) are 1 for \(x^2\), -1 for \(x\), and -2 as the constant term, all of which are real numbers.
This is a crucial concept in polynomial equations and their factorizations. For example, when a polynomial is expressed as \(p(x) = (x-2)(x+1)\), the coefficients of \(x^2-x-2\) are 1 for \(x^2\), -1 for \(x\), and -2 as the constant term, all of which are real numbers.
- Real coefficients ensure that the solutions (when solving equations) are real numbers.
- For a polynomial with real coefficients, complex roots appear in conjugate pairs.
Quadratic Polynomial
Quadratic polynomials are polynomials of degree 2, which means the highest exponent of the variable \(x\) is 2. The general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients with \(a eq 0\). These are some of the simplest forms of polynomials and play a significant role in various mathematical domains.
A quadratic polynomial can have at most two zeros or roots since its degree is 2. These roots can be real or complex. In our example, constructing a quadratic polynomial with zeros at \(x = 2\) and \(x = -1\), we express it as \(p(x) = (x-2)(x+1)\). Simplifying this gives \(p(x) = x^2-x-2\).
A quadratic polynomial can have at most two zeros or roots since its degree is 2. These roots can be real or complex. In our example, constructing a quadratic polynomial with zeros at \(x = 2\) and \(x = -1\), we express it as \(p(x) = (x-2)(x+1)\). Simplifying this gives \(p(x) = x^2-x-2\).
- Quadratic polynomials form a symmetric parabola when graphed.
- The vertex of the parabola is a significant point which is the maximum or minimum of the graph.
- The axis of symmetry of a parabola given by a quadratic \(ax^2 + bx + c\) is \(x = -\frac{b}{2a}\).
Other exercises in this chapter
Problem 39
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