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Magazine Chapter 4: Problem 3
If \(c\) is a zero of the polynomial \(P,\) which of the following statements must be true? (a) \(P(c)=0\) (b) \(P(0)=c\) (c) \(x-c\) is a factor of \(P(x)\) . (d) \(c\) is the \(y\) -intercept of the graph of \(P\) .
Short Answer
Expert verified
Only statements (a) and (c) are true.
Step by step solution
01
Understanding Zero of a Polynomial
A zero of the polynomial \(P\), denoted as \(c\), is a value for which the polynomial evaluates to zero. By definition, if \(c\) is a zero, then \(P(c) = 0\). This corresponds to statement (a). Hence, (a) must be true.
02
Evaluating Statement (b)
Statement (b), \(P(0) = c\), suggests \(c\) as the result of the polynomial evaluated at \(x = 0\). This is incorrect for zeros unless \(c = 0\). Generally, this statement does not hold true for a zero \(c\) of a polynomial.
03
Factor Theorem Application for Statement (c)
The factor theorem states that \(x-c\) is a factor of \(P(x)\) if and only if \(c\) is a zero of \(P(x)\). Therefore, \(c\) being a zero directly implies that \(x-c\) is a factor. Hence, statement (c) must be true.
04
Analyzing Statement (d)
Statement (d) claims \(c\) is the \(y\)-intercept of the graph. The \(y\)-intercept of a polynomial is found by evaluating \(P(0)\), which may not necessarily be equal to \(c\). Therefore, this statement is generally false unless specific conditions apply.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. These expressions are handled using operations like addition, subtraction, multiplication, and non-negative integer exponentiation of variables. The structure of a polynomial can vary in complexity, from very simple to quite complicated forms. Here are a few things to keep in mind:
- The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial \(5x^3 - 4x^2 + 2x - 1\), the degree is 3.
- Polynomials can be classified based on their degree. For example, a linear polynomial has a degree of 1, a quadratic has a degree of 2, and a cubic has a degree of 3.
- The coefficients of a polynomial are the numbers that are constants or that multiply the variables. They play a crucial role in determining the shape and behavior of the polynomial's graph.
Factor Theorem
The Factor Theorem provides a simple yet powerful way to link the roots of a polynomial to its factors. It's an extension of the concept that if you divide something and it leaves no remainder, then it fits perfectly. Here鈥檚 how it works:
- If \(c\) is a zero of the polynomial \(P(x)\), then \(x-c\) is a factor of \(P(x)\). This essentially means that if you substitute \(c\) into the polynomial and get zero \(P(c) = 0\), then the polynomial can be rewritten to include \(x-c\) as a component.
- The theorem is incredibly useful for factoring polynomials, as it allows us to "break down" the polynomial into simpler components that multiply to give the original expression.
- It is also instrumental in finding the roots of higher-degree polynomials, where direct computation is often cumbersome.
Polynomial Graph Analysis
Analyzing a polynomial graph gives us a visual representation of solutions and behaviors of the polynomial equation. The graph can tell us a lot about the polynomial, including where it touches or crosses the x-axis. Here are some key aspects:
- The x-intercepts of the graph are the zeros of the polynomial, meaning where the polynomial equals zero. These are solutions to the equation \(P(x) = 0\).
- The degree of the polynomial can often be deduced from the number of turns or "wiggles" in the graph. A higher degree polynomial will typically have more potential turns.
- The y-intercept is the point where the graph crosses the y-axis, which is found by evaluating \(P(0)\). This is not to be confused with the zeros unless the zero is at \(x=0\).
Roots of a Polynomial
Roots of a polynomial, often called zeros, are values of \(x\) that satisfy the equation \(P(x) = 0\). Finding these roots is a fundamental part of solving polynomial equations. Here are some essential points:
- Roots can be real or complex numbers. Real roots are the x-values where the graph crosses the x-axis, while complex roots do not show on the standard Cartesian plane but are equally important.
- The number of roots of a polynomial is equal to its degree. For instance, a quadratic (degree 2) polynomial can have up to two roots.
- Roots serve as vital reference points when factoring polynomials and can be determined using methods like the quadratic formula, factoring, or graph analysis.
- Finding all possible rational roots of a polynomial is often approached using the Rational Root Theorem before confirming them using substitution into the polynomial.
