The fundamental concept of Polynomial & Rational Function. <\/p>\n<\/li>\n<\/ul>\n
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Degree & Leading coefficient, Types of Infinity. <\/p>\n<\/li>\n<\/ul>\n
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Zeros & Factors, Graphical Behavior <\/p>\n<\/li>\n<\/ul>\n
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Types of Asymptotes, Application & Example <\/p>\n<\/li>\n<\/ul>\n
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Fundamental Concept in Polynomial & Rational Functions<\/h2>\n
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A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication operations. <\/p>\n<\/li>\n<\/ul>\n
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A rational function is a fraction of two polynomials, where the numerator and denominator are both polynomials. <\/p>\n<\/li>\n<\/ul>\n
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The degree of a polynomial is determined by the highest power of the variable present in the expression. For example, a polynomial of degree 3 would have a term with a variable raised to the power of 3. <\/p>\n<\/li>\n<\/ul>\n
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The degree of a rational function is determined by the highest degree of the numerator or the denominator polynomial, whichever is larger. <\/p>\n<\/li>\n<\/ul>\n
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A polynomial of degree 7 has terms with the variable raised to powers from 0 to 7, with each power represented by a coefficient. <\/p>\n<\/li>\n<\/ul>\n
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A rational function with a polynomial of degree 7 in the numerator and a polynomial of degree 7 in the denominator is called a polynomial over polynomial function of degree 7. <\/p>\n<\/li>\n<\/ul>\n
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The graph of a polynomial over polynomial function of degree 7 can exhibit up to 7 turning points, where the direction of the curve changes from increasing to decreasing or vice versa. <\/p>\n<\/li>\n<\/ul>\n
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The leading term of a polynomial over polynomial function of degree 7 is the term with the highest power of the variable. It determines the behavior of the function as the input approaches positive or negative infinity. <\/p>\n<\/li>\n<\/ul>\n
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The vertical asymptotes of a rational function occur at values of the variable where the denominator polynomial is equal to zero but the numerator polynomial is not zero. <\/p>\n<\/li>\n<\/ul>\n
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The horizontal asymptote of a rational function is determined by the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator polynomials. <\/p>\n<\/li>\n<\/ul>\n
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The x-intercepts of a polynomial over polynomial function of degree 7 occur at the values of the variable where the numerator polynomial is equal to zero. <\/p>\n<\/li>\n<\/ul>\n
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The behavior of a polynomial over polynomial function of degree 7 near its x-intercepts depends on the multiplicity of each root. If a root has an odd multiplicity, the graph crosses the x-axis at that point. If a root has an even multiplicity, the graph touches but does not cross the x-axis at that point. <\/p>\n<\/li>\n<\/ul>\n
Additional Concept for Polynomial Function <\/strong><\/h2>\n
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- The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots (including repeated roots) when counted with multiplicity. <\/li>\n
- The Remainder Theorem states that if a polynomial f(x) is divided by (x – a), the remainder is equal to f(a). This theorem is helpful in evaluating polynomials at specific values. <\/li>\n
- The Factor Theorem states that if a polynomial f(a) is equal to zero, then (x – a) is a factor of the polynomial. This theorem helps in finding the factors and roots of a polynomial. <\/li>\n
- The Intermediate Value Theorem states that if a polynomial function is continuous on an interval [a, b], and f(a) and f(b) have opposite signs, then there exists at least one root of the polynomial in the interval (a, b). This theorem is useful for locating the roots of a polynomial. <\/li>\n
- The Leading Coefficient Test is used to determine the end behavior of a polynomial function. If the leading coefficient is positive, the graph of the polynomial rises to the right and falls to the left. If the leading coefficient is negative, the graph does the opposite. <\/li>\n
- The Division Algorithm allows us to divide polynomials by long division or synthetic division to obtain quotient and remainder expressions. <\/li>\n
- The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x) on an interval [a, b], then the definite integral of f(x) from a to b is equal to F(b) – F(a). This theorem is particularly useful for finding areas under polynomial curves. <\/li>\n
- Rolle's Theorem states that if a polynomial function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0. This theorem provides conditions for the existence of critical points in a polynomial function. <\/li>\n
- The Factor Theorem can be used in reverse to express a polynomial as a product of its linear factors, which can help in factoring higher-degree polynomials. <\/li>\n
- The polynomial long division algorithm is used to divide polynomials when the divisor is not a linear factor. This method allows for finding the quotient and remainder. <\/li>\n
- The Descartes' Rule of Signs helps in determining the number of positive or negative roots of a polynomial by examining sign changes in its coefficients. <\/li>\n
- Polynomial functions can exhibit symmetry. Even-degree polynomials (with all even powers) are symmetric about the y-axis, while odd-degree polynomials (with at least one odd power) are symmetric about the origin (the point (0,0)). <\/li>\n<\/ol>\n
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<\/p>\n(Polynomial Factor) <\/p>\n
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Properties of Polynomial & Rational Function: – <\/strong><\/h1>\n
Polynomial Functions: <\/strong><\/h2>\n
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Polynomial functions are continuous for all real numbers. This means that there are no breaks, holes, or jumps in their graphs. <\/p>\n<\/li>\n<\/ul>\n
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Polynomial functions have even degrees or odd degrees, which determine the end behavior. Even-degree polynomials tend to rise or fall indefinitely, while odd-degree polynomials have opposite end behavior. <\/p>\n<\/li>\n<\/ul>\n
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The number of turning points in a polynomial function is at most equal to its degree minus one. Turning points are where the graph changes direction from increasing to decreasing or vice versa. <\/p>\n<\/li>\n<\/ul>\n
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The graph of a polynomial function may have x-intercepts, where the function crosses the x-axis. The number of x-intercepts is at most equal to the degree of the polynomial. <\/p>\n<\/li>\n<\/ul>\n
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The graph of a polynomial function may have y-intercepts, where the function crosses the y-axis. The y-intercept is the value of the function when x is zero. <\/p>\n<\/li>\n<\/ul>\n
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Rational Functions: <\/strong><\/h2>\n
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Rational functions can have vertical asymptotes, which are vertical lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator of the rational function is equal to zero. <\/p>\n<\/li>\n<\/ul>\n
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Rational functions can have horizontal asymptotes, which are horizontal lines that the graph approaches as x approaches positive or negative infinity. The presence and position of horizontal asymptotes depend on the degrees of the numerator and denominator polynomials. <\/p>\n<\/li>\n<\/ul>\n
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Rational functions can have holes or removable discontinuities. These occur when both the numerator and denominator have common factors that cancel out, resulting in a hole in the graph. <\/p>\n<\/li>\n<\/ul>\n
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The graph of a rational function may have x-intercepts, which are the values of x where the function equals zero. X-intercepts occur where the numerator of the rational function is equal to zero. <\/p>\n<\/li>\n<\/ul>\n
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The graph of a rational function may have y-intercepts, which are the values of y where x is equal to zero. The y-intercept is the value of the function when x is zero. <\/p>\n<\/li>\n<\/ul>\n
<\/p>\n
Degree & Leading Coefficient <\/strong><\/h2>\n
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Degree of a Polynomial: <\/strong><\/h3>\n<\/li>\n<\/ol>\n
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The degree of a polynomial is the highest power of the variable in the polynomial expression. <\/p>\n<\/li>\n<\/ul>\n
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The degree helps determine the behavior and shape of the polynomial function. <\/p>\n<\/li>\n<\/ul>\n
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For example, a polynomial of degree 3 is called a cubic polynomial, while a polynomial of degree 2 is called a quadratic polynomial. <\/p>\n<\/li>\n<\/ul>\n
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Leading Term and Leading Coefficient: <\/p>\n<\/li>\n<\/ol>\n
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The leading term of a polynomial is the term with the highest power of the variable. <\/p>\n<\/li>\n<\/ul>\n
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The leading coefficient is the coefficient of the leading term. <\/p>\n<\/li>\n<\/ul>\n
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The leading term and leading coefficient play a crucial role in determining the end behavior of a polynomial function. <\/p>\n<\/li>\n<\/ul>\n
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The sign of the leading coefficient influences whether the polynomial rises or falls as the input approaches positive or negative infinity. <\/p>\n<\/li>\n<\/ul>\n
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Relationship between Degree and Turning Points: <\/p>\n<\/li>\n<\/ol>\n
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The number of turning points in a polynomial function is at most equal to the degree of the polynomial minus one. <\/p>\n<\/li>\n<\/ul>\n
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A turning point is where the graph of the polynomial changes direction from increasing to decreasing or vice versa. <\/p>\n<\/li>\n<\/ul>\n
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For example, a quadratic polynomial of degree 2 can have at most one turning point, while a cubic polynomial of degree 3 can have at most two turning points. <\/p>\n<\/li>\n<\/ul>\n
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Relationship between Degree and x-Intercepts: <\/p>\n<\/li>\n<\/ol>\n
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The number of x-intercepts (also known as roots or zeros) of a polynomial function is at most equal to its degree. <\/p>\n<\/li>\n<\/ul>\n
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An x-intercept is a value of x where the polynomial equals zero, causing the graph to cross the x-axis. <\/p>\n<\/li>\n<\/ul>\n
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For example, a quadratic polynomial of degree 2 can have at most two x-intercepts, while a cubic polynomial of degree 3 can have at most three x-intercepts. <\/p>\n<\/li>\n<\/ul>\n
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Relationship between Leading Coefficient and End Behavior: <\/p>\n<\/li>\n<\/ol>\n
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The leading coefficient of a polynomial function affects the end behavior of the graph. <\/p>\n<\/li>\n<\/ul>\n
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If the leading coefficient is positive, the polynomial rises on both ends as the input approaches positive and negative infinity. <\/p>\n<\/li>\n<\/ul>\n
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If the leading coefficient is negative, the polynomial falls on both ends as the input approaches positive and negative infinity. <\/p>\n<\/li>\n<\/ul>\n
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The leading coefficient determines the overall direction of the graph at the far ends of the x-axis. <\/p>\n<\/li>\n<\/ul>\n
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<\/p>\n<\/p>\n
Example: <\/p>\n
Factor a given polynomial f(x) = x3<\/sup> – 4x2<\/sup> – 11x + 30. <\/strong><\/p>\n
Solution: <\/strong><\/p>\n
To factor the given polynomial, we can start by checking for any possible rational roots using the Rational Root Theorem. According to the theorem, the possible rational roots are the factors of the constant term (30) divided by the factors of the leading coefficient (1). <\/p>\n
The factors of 30 are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30. <\/p>\n
<\/p>\n
We can now test these values by substituting them into the polynomial to see if they are roots. We'll start with the smallest possible value, -1: <\/p>\n
<\/p>\n
f(-1) = (-1)3<\/sup> – 4(-1)2<\/sup> – 11(-1) + 30 <\/p>\n
= -1 + 4 + 11 + 30 <\/p>\n
= 44 <\/p>\n
<\/p>\n
Since f(-1) ≠ 0, -1 is not a root of the polynomial. <\/p>\n
<\/p>\n
We continue this process, testing the remaining possible rational roots. By trying out different values, we find that x = 3 is a root of the polynomial, which means (x – 3) is a factor. <\/p>\n
Using polynomial long division or synthetic division, we can divide the original polynomial f(x) by (x – 3): <\/p>\n
The quotient obtained is x2 – x – 10. Now we have factored the polynomial as follows: <\/p>\n
f(x) = (x – 3)(x2<\/sup> – x – 10) <\/p>\n
To further factor the quadratic factor, we can solve for its roots using factoring, completing the square, or applying the quadratic formula: <\/p>\n
X2<\/sup> – x – 10 = 0 <\/p>\n
(x – 5)(x + 2) = 0 <\/p>\n
Therefore, the quadratic factor x2 – x – 10 can be factored as (x – 5)(x + 2). <\/p>\n
Thus, the fully factored form of the polynomial f(x) is: <\/p>\n
f(x) = (x – 3)(x – 5)(x + 2). <\/p>\n
This example demonstrates the process of factoring a given polynomial using rational root testing and long division\/synthetic division to find the linear factors. <\/p>\n
<\/p>\n
Key Points <\/strong><\/p>\n
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A polynomial factor of a polynomial function is a linear expression that divides the polynomial evenly, leaving no remainder. <\/p>\n<\/li>\n<\/ul>\n
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The factor theorem states that if (x – a) is a factor of a polynomial, then plugging in a into the polynomial will result in zero. <\/p>\n<\/li>\n<\/ul>\n
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Polynomial factors help in finding the x-intercepts or roots of a polynomial function. <\/p>\n<\/li>\n<\/ul>\n
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The degree of a polynomial factor is always 1 since it is a linear expression. <\/p>\n<\/li>\n<\/ul>\n
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The Fundamental Theorem of Algebra states that every polynomial of degree greater than 0 has at least one complex root. <\/p>\n<\/li>\n<\/ul>\n
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If a polynomial has repeated factors, the corresponding root will have multiplicity greater than 1. <\/p>\n<\/li>\n<\/ul>\n
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Factoring a polynomial can simplify complex expressions and reveal key characteristics of the polynomial function. <\/p>\n<\/li>\n<\/ul>\n
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Long division or synthetic division can be used to divide a polynomial by a linear factor and obtain the quotient polynomial. <\/p>\n<\/li>\n<\/ul>\n
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Rational factors help in finding the vertical asymptotes and holes in a rational function. <\/p>\n<\/li>\n<\/ul>\n
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Holes occur when a common factor cancels out both the numerator and denominator of the rational function. <\/p>\n<\/li>\n<\/ul>\n
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Vertical asymptotes occur where the denominator polynomial becomes zero but the numerator polynomial does not. <\/p>\n<\/li>\n<\/ul>\n
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The behavior of a rational function near its vertical asymptotes can be determined by examining the sign of the numerator and denominator polynomials. <\/p>\n<\/li>\n<\/ul>\n
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Horizontal asymptotes of a rational function are determined by the degrees of the numerator and denominator polynomials. <\/p>\n<\/li>\n<\/ul>\n
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Rational factors aid in simplifying rational functions, allowing for easier analysis and graphing. <\/p>\n<\/li>\n<\/ul>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
Unit: Polynomial & Rational Function Chapter: Change of Quantity & Behaviour of Functions Reference: – Degree coefficient, Leading coefficient, Polynomial function, Behavior of function, Positive infinity, Negative infinity, Zeros & Factors, Multiplicity of zeros, Turning points, Graphical Behavior, Rational functions, Vertical Asymptotes, Horizontal Asymptotes, Oblique Asymptotes After studying this chapter, you should be able […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[628],"tags":[],"class_list":["post-9178","post","type-post","status-publish","format-standard","hentry","category-ap-precalculus"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9178","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9178"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9178\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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