Polynomial Functions and Their Properties

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As x approaches positive or negative infinity, the function approaches positive infinity.

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<p>Analyze the sign of the derivative around the point.</p> Signup and view all the answers

<p>Domain is the set of all possible input values, while range is the set of all possible output values.</p> Signup and view all the answers

<p>Multiplicity indicates how many times a root appears in the factored form of the polynomial. Even multiplicity creates a 'bounce' at the x-axis, while odd multiplicity creates a 'cross' at the x-axis.</p> Signup and view all the answers

<p>The end behavior will be flipped.</p> Signup and view all the answers

<p>The degree of a polynomial function indicates the maximum number of real roots it can have.</p> Signup and view all the answers

<p>Set each factor equal to zero and solve for x.</p> Signup and view all the answers

<p>The maximum number of turning points is one less than the degree.</p> Signup and view all the answers

<p>The Rational Root Theorem states that any rational root of a polynomial must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is -12 and the leading coefficient is 10. The factors of -12 are ±1, ±2, ±3, ±4, ±6, and ±12. The factors of 10 are ±1, ±2, ±5, and ±10. Therefore, the possible rational roots are ±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1, ±1/2, ±2/2, ±3/2, ±4/2, ±6/2, ±12/2, ±1/5, ±2/5, ±3/5, ±4/5, ±6/5, ±12/5, ±1/10, ±2/10, ±3/10, ±4/10, ±6/10, and ±12/10.</p> Signup and view all the answers

<p>The roots of a polynomial are the values of x that make the polynomial equal to zero. The x-intercepts of the graph of a polynomial are the points where the graph crosses the x-axis. A root of a polynomial is an x-intercept if and only if the root is a real number. Complex roots do not correspond to x-intercepts on the graph of the function because they do not intersect the x-axis. Therefore, all of the roots of a polynomial are x-intercepts only if all of the roots are real numbers. If a polynomial has complex roots, then those roots will not correspond to x-intercepts on the graph.</p> Signup and view all the answers

<p>The Factor Theorem states that a binomial x - a is a factor of a polynomial f(x) if and only if f(a) = 0. To determine if a binomial is a factor of a polynomial, substitute the value of a into the polynomial and evaluate. If the result is zero, then the binomial is a factor of the polynomial. For example, to determine if x - 4 is a factor of the polynomial <code>f(x) = 2x^3 - 7x^2 - 10x + 24</code>, we substitute x = 4 into the polynomial and evaluate: f(4) = 2(4)^3 - 7(4)^2 - 10(4) + 24 = 128 - 112 - 40 + 24 = 0. Since f(4) = 0, we can conclude that x - 4 is a factor of <code>f(x) = 2x^3 - 7x^2 - 10x + 24</code>.</p> Signup and view all the answers

<p>The Fundamental Theorem of Algebra states that every polynomial equation with complex coefficients has at least one complex root. This implies that a polynomial of degree n has exactly n roots, counting multiplicity. Therefore, a degree 7 polynomial will have exactly 7 roots.</p> Signup and view all the answers

<p>The multiplicity of a root is the number of times that a root appears in the factorization of a polynomial. For example, if the root x = 2 appears in the factorization of a polynomial three times, then the multiplicity of the root x = 2 is 3. The multiplicity of a root affects the graph of the polynomial at the corresponding x-intercept. If the multiplicity of a root is odd, then the graph will cross the x-axis at that point. If the multiplicity of a root is even, then the graph will touch the x-axis at that point but will not cross it.</p> Signup and view all the answers

<p>To write a polynomial function in standard form with a given set of zeros and a specified leading coefficient, follow these steps: 1. Write the linear factors corresponding to each zero. 2. Multiply the linear factors together to obtain the polynomial in factored form. 3. Expand the factored form to obtain the polynomial in standard form. 4. Multiply the polynomial by the specified leading coefficient.</p> Signup and view all the answers

<p>Complex roots of a polynomial always come in conjugate pairs because the coefficients of a polynomial with real coefficients are also real. The conjugate of a complex number <code>a + bi</code> is <code>a - bi</code>. If <code>a + bi</code> is a root of a polynomial with real coefficients, then its conjugate, <code>a - bi</code>, must also be a root of the polynomial. For example, if the polynomial <code>f(x) = x^2 - 2x + 5</code> has the root <code>1 + 2i</code>, then its conjugate, <code>1 - 2i</code>, must also be a root of the polynomial. This is because the polynomial has real coefficients: <code>x^2 - 2x + 5</code> = <code>(x - (1 + 2i))(x - (1 - 2i))</code>.</p> Signup and view all the answers

<p>The degree of a polynomial provides information about its end behavior and the maximum number of turning points. The end behavior of a polynomial is determined by the sign of the leading coefficient and the degree of the polynomial. If the degree of the polynomial is even and the leading coefficient is positive, then the graph will rise both on the left and on the right. If the degree of the polynomial is even and the leading coefficient is negative, then the graph will fall both on the left and on the right. If the degree of the polynomial is odd and the leading coefficient is positive, then the graph will rise on the right and fall on the left. If the degree of the polynomial is odd and the leading coefficient is negative, then the graph will fall on the right and rise on the left. The maximum number of turning points of a polynomial is one less than its degree. For example, a polynomial of degree 3 would have a maximum of 2 turning points. A polynomial of degree 4 would have a maximum of 3 turning points.</p> Signup and view all the answers

<p>It is impossible to have a cubic function with two of the roots being 2i and √3 because complex roots of polynomials with real coefficients must come in conjugate pairs. Since <code>2i</code> is a root, its conjugate, <code>-2i</code>, must also be a root. Therefore, a cubic function cannot have only <code>2i</code> and √3 as roots, as the polynomial must have at least one more root.</p> Signup and view all the answers

<p>The Intermediate Value Theorem states that if a polynomial function f(x) is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there exists at least one root of the polynomial within the interval [a, b]. To apply this theorem to find roots, we can evaluate the polynomial function at two points, a and b, within the interval of interest. If the function values at a and b have opposite signs, then we know that there is a root within the interval. By narrowing down the interval, we can approximate the location of the root. This method, while not precise, allows us to find roots through an iterative process of evaluation and interval reduction.</p> Signup and view all the answers

An expression made of variables and coefficients using only addition, subtraction, and multiplication.

A representation of a polynomial using its zeros, expressed as f(x) = a(x-r1)(x-r2)...(x-rn).

The behavior of a polynomial's graph as x approaches infinity or negative infinity.

Ranges of x-values where the function's value is rising.

Ranges of x-values where the function's value is falling.

Points where a polynomial changes from increasing to decreasing or vice versa.

The highest point in a specific interval of a polynomial graph.

The lowest point in a specific interval of a polynomial graph.

The highest exponent of the variable in a polynomial expression.

The coefficient of the term with the highest degree in a polynomial.

A polynomial that represents the volume of a box after cutting corners.

A polynomial expression that divides another polynomial without a remainder.

Values that can be tested as roots of a polynomial function.

Values of x where the polynomial equals zero.

The number of times a root appears in a polynomial.

Roots that include imaginary numbers in polynomial equations.

A format that expresses a polynomial based on roots x-intercepts.

The solutions where a cubic equation crosses the x-axis.

How the shape of the graph changes based on degree and roots.