Chapter 2.7 and 7.3
Chapter 2 Polynomial and Rational Functions 2.1 Complex Numbers 2.2 Quadratic Functions 2.3 Polynomial Functions and Their Graphs 2.4 Dividing Polynomials; Remainder and Factor Theorems 2.5 Zeros of Polynomial Functions 2.6 Rational Functions and Their Graphs 2.7 Polynomial and Rational Inequalities 2.8 Modeling Using Variation
2.7 Polynomial and Rational Inequalities Polynomial Inequality A polynomial inequality can be expressed as f x < 0, f x > 0, f x 0, or f x 0, where f is a polynomial function. Rational Inequality A rational inequality can be expressed as f x < 0, f x > 0, f x 0, or f x 0, where f is a rational function. Example: A car s required stopping distance, f x, in feet, on dry pavement traveling at xmiles per hour can be modeled by the quadratic function f x = 0.0875x 2 0.4x + 66.6. How can we use this function to determine speeds on dry pavement requiring stopping distance that exceed the length of one and one-half football field, or 540 feet? Then we must solve the inequality f x = 0.0875x 2 0.4x + 66.6 > 540.
2.7 Polynomial and Rational Inequalities Procedure of Solving Polynomial Inequalities 1. Express the inequality in the form f x < 0 or f x > 0, where f is a polynomial function. 2. Solve the equation f x = 0. The real solutions are the boundary points. 3. Locate these boundary points on a number line, thereby dividing the number line into intervals. 4. Choose one representative number, called a test value, within each interval and evaluate f at that number. If the value of f is positive, then f x > 0 for all numbers, x, in the interval. If the value of f is negative, then f x < 0 for all numbers, x, in the interval. 5. Write the solution set, selecting the interval or intervals that satisfy the given inequality.
2.7 Polynomial and Rational Inequalities Example 1. Solve and graph the solution set on a real number line: 2x 2 + x > 15.
2.7 Polynomial and Rational Inequalities Example 2. Solve and graph the solution set on a real number line: 4x 2 1 2x.
2.7 Polynomial and Rational Inequalities Example 3. Solve and graph the solution set on a real number line: x 3 + x 2 4x + 4.
2.7 Polynomial and Rational Inequalities Procedure of Solving Rational Inequalities 1. Bring all terms to one side, obtaining zero on the other side. 2. Express the rational function on the nonzero side as a single quotient. 3. Set the numerator and the denominator of the rational function f equal to zero. The solutions of these equations serve as boundary points that separate the real number line into intervals. At this point, the procedure is the same as the one we used for solving polynomial inequalities. Here, zeros from the denominator should NOT be included in the solution set.
2.7 Polynomial and Rational Inequalities Example 4. Solve and graph the solution set on a real number line: x + 1 x + 3 2.
2.7 Polynomial and Rational Inequalities Example 5. (Application) A ball is thrown vertically upward from the top of the Learning Tower of Pisa (190 feet high) with an initial velocity of 96 feet per second. During which time period will the ball s height exceed that of the tower? Use the position function s t = 16t 2 + v 0 t + s 0 (v 0 = initial velocity, s 0 = initial position) for a free-falling object. End of 2.7
Chapter 7 Systems of Equations and Inequalities 7.1 Systems of Linear Equations in Two Variables 7.2 Systems of Linear Equations in Three Variables 7.4 Systems of Nonlinear Equations in Two Variables 7.5 Systems of Inequalities 7.6 Linear Programming
Partial Fraction Decompositions We know how to use common denominators to write a sum or difference of rational expressions as a single rational expression. For example, 3 x 4 2 x + 2 = 3 x 4 x + 2 x + 2 2 x + 2 x 4 3 x + 2 2 x 4 = x 4 x 4 x + 2 3x + 6 2x + 8 = x 4 x + 2 = x + 14 x 4 x + 2. This procedure in used to simplify multiple rational expressions. In mathematics, it is also important to know the original multiple rational expressions when a simplified single rational expression is given. Each of the original multiple rational expressions is called a partial fraction. The sum of these fractions is called the partial fraction decomposition of the given single rational expression.
Partial Fraction Decompositions Partial fraction decompositions can be written for rational expressions of the form P x Q x, where P and Q have no common factors and the highest power in the numerator is less than the highest power in the denominator. If the highest power in the numerator is greater than or equal to the highest power in the denominator, we need to perform a long division first. 9x 2 9x + 6 2x 1 x + 2 x 2, 5x 3 3x 2 + 7x 3 x 2 + 1 x 2 + 1 2, x 1 x + 1 Theorem Every polynomial with real coefficients can be factored as products of linear factors and prime (or irreducible) quadratic factors. (A prime (pr irreducible) quadratic polynomial is a polynomial which cannot be factored in real numbers.)
Partial Fraction Decompositions: Four Different Types Partial fraction decompositions of a rational expression depends on the factors of the denominator. We consider four cases involving different kinds of factors in the denominator. 1. The denominator is a product of distinct linear factors. 2. The denominator is a product of linear factors, some of which are repeated. 3. The denominator has prime quadratic factors, none of which is repeated. 4. The denominator has a repeated prime quadratic factors.
Partial Fraction Decompositions: Type 1 The denominator is a product of distinct linear factors. The form of the partial fraction decomposition for a rational expression with distinct linear factors in the denominator is P x a 1 x + b 1 a 2 x + b 2 a n x + b n = A 1 a 1 x + b 1 + A 2 a 2 x + b 2 + + for some constants A 1, A 2,, A n. The equation is true for all x. A n a n x + b n
Steps Partial Fraction Decompositions 1. Set up the partial fraction decomposition with the unknown constants A, B, C, etc., in the numerators of the decomposition. 2. Multiply both sides of the resulting equation by the least common denominator. 3. Simplify the right side of the eqution. 4. Write both sides in descending powers, equate coefficients of like powers of x, and equate constant term. 5. Solve the resulting linear system for A, B, C, etc. 6. Substitute the value for A, B, C, etc., into the equation in step 1 and write the partial fraction decomposition. Example 1. Find the partial fraction decomposition of x + 14 (x 4)(x + 2).
Partial Fraction Decompositions: Type 2 The denominator is a product of linear factors, some of which are repeated. The form of the partial fraction decomposition for a rational expression with repeated linear factors in the denominator is P x ax + b n = A 1 ax + b + A 2 ax + b 2 + + A n ax + b n for some constants A 1, A 2,, A n. The equation is true for all x. Example 2. Find the partial fraction decomposition of x 18 x x 3 2.
Partial Fraction Decompositions: Type 3 The denominator has prime quadratic factors, none of which is repeated. If the denominator of a rational expression has a nonrepeated prime quadratic factor ax 2 + bx + c b 2 4ac < 0, then the partial fraction decomposition will contain a term of the form Ax + B for some constants A and B. ax 2 + bx + c Example 3. Find the partial fraction decomposition of 3x 2 + 17x + 14 (x 2)(x 2 + 2x + 4).
Partial Fraction Decompositions: Type 4 The denominator has a repeated prime quadratic factors. The FORM of the partial fraction decomposition for a rational expression containing the prime factor ax 2 + bx + c b 2 4ac < 0 occurring n times as its denominator is P(x) ax 2 + bx + c n = A 1x + B 1 ax 2 + bx + c + A 2 x + B 2 ax 2 + bx + c 2 + + A n x + B n ax 2 + bx + c n for some constants A 1, A 2, A n, B 1, B 2,, B n. The equation is true for all x. Example 4. Find the partial fraction decomposition of 5x 3 3x 2 + 7x 3 x 2 + 1 2.
Partial Fraction Decompositions: What if the degree of the numerator is greater than or equal to the degree of the denominator? - A long division can be used to write as P x r x = q x + Q x Q x, where the degree of r(x) is less than the degree of Q(x). Then perform the partial fraction decomposition procedure for r x Q x. Example 4. Find the partial fraction decomposition of x 2 + 1 x 2 1 = x 2 + 1 x 1 x + 1 End of 7.3