Explores the connection between the zeros of a polynomial function, the roots of a polynomial equation, and the factors of the polynomial. Introduces the Factor Theorem and Remainder Theorem.
What is a 'zero' of a polynomial function \(P(x)\)?
A value of \(x\) for which \(P(x) = 0\). These are the \(x\)-values where the function's graph intersects the \(x\)-axis.
Explain the relationship between the terms: 'zeros' of a function, 'roots' of an equation, and 'x-intercepts' of a graph.
They are largely equivalent concepts. If \(k\) is a zero of the function \(P(x)\), then \(k\) is a root of the equation \(P(x)=0\), and \((k, 0)\) is an x-intercept of the graph of \(y=P(x)\).
What does the Remainder Theorem state?
If a polynomial \(P(x)\) is divided by a linear factor \((x - k)\), the remainder is equal to the value of the function at \(k\). In other words, Remainder = \(P(k)\).
Use the Remainder Theorem to find the remainder when \(P(x) = x^3 - 4x^2 + 2x + 5\) is divided by \((x - 3)\).
According to the Remainder Theorem, the remainder is \(P(3)\). \(P(3) = (3)^3 - 4(3)^2 + 2(3) + 5 = 27 - 36 + 6 + 5 = 2\). The remainder is \(2\).
What does the Factor Theorem state?
A number \(k\) is a zero of a polynomial \(P(x)\) if and only if \((x - k)\) is a factor of \(P(x)\). This is a direct consequence of the Remainder Theorem, where the remainder is \(0\).
How can you use the Factor Theorem to determine if \((x + 2)\) is a factor of \(P(x) = x^3 + 8\)?
According to the Factor Theorem, \((x + 2)\) is a factor if \(P(-2) = 0\). \(P(-2) = (-2)^3 + 8 = -8 + 8 = 0\). Since the result is \(0\), \((x + 2)\) is a factor.
If a polynomial has zeros at \(x = -1\), \(x = 2\), and \(x = 5\), what is a possible factored form of the polynomial?
Using the Factor Theorem, if \(k\) is a zero, then \((x - k)\) is a factor. So, the factors are \((x - (-1))\), \((x - 2)\), and \((x - 5)\). A possible polynomial is \(P(x) = a(x+1)(x-2)(x-5)\), where \(a\) is any non-zero constant.
Find a third-degree polynomial function with zeros at \(-3\), \(0\), and \(4\).
The factors are \((x - (-3))\), \((x - 0)\), and \((x - 4)\), which are \((x+3)\), \(x\), and \((x-4)\). Multiplying them gives: \(P(x) = x(x+3)(x-4) = x(x^2 - x - 12) = x^3 - x^2 - 12x\).
What is the relationship between the degree of a polynomial and the maximum number of 'turning points' on its graph?
A polynomial of degree \(n\) has at most \(n-1\) turning points (places where the graph changes from increasing to decreasing or vice versa). For example, a cubic (degree \(3\)) has at most \(2\) turning points.
