Remainder theorem examples with answers

2 Equations and Inequalities • Introduction to Equations and Inequalities • 2.1 The Rectangular Coordinate Systems and Graphs • 2.2 Linear Equations in One Variable • 2.3 Models and Applications • 2.4 Complex Numbers • 2.5 Quadratic Equations • 2.6 Other Types of Equations • 2.7 Linear Inequalities and Absolute Value Inequalities • 5 Polynomial and Rational Functions • Introduction to Polynomial and Rational Functions • 5.1 Quadratic Functions • 5.2 Power Functions and Polynomial Functions • 5.3 Graphs of Polynomial Functions • 5.4 Dividing Polynomials • 5.5 Zeros of Polynomial Functions • 5.6 Rational Functions • 5.7 Inverses and Radical Functions • 5.8 Modeling Using Variation • 6 Exponential and Logarithmic Functions • Introduction to Exponential and Logarithmic Functions • 6.1 Exponential Functions • 6.2 Graphs of Exponential Functions • 6.3 Logarithmic Functions • 6.4 Graphs of Logarithmic Functions • 6.5 Logarithmic Properties • 6.6 Exponential and Logarithmic Equations • 6.7 Exponential and Logarithmic Models • 6.8 Fitting Exponential Models to Data • 7 Systems of Equations and Inequalities • Introduction to Systems of Equations and Inequalities • 7.1 Systems of Linear Equations: Two Variables • 7.2 Systems of Linear Equations: Three Variables • 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables • 7.4 Partial Fractions • 7.5 Matrices and Matrix Operations • 7.6 Solving Systems with Gaussian Elimination • 7.7 Solving Systems with Inverses • 7.8 Solvin...

2 Solving Linear Equations • Introduction • 2.1 Use a General Strategy to Solve Linear Equations • 2.2 Use a Problem Solving Strategy • 2.3 Solve a Formula for a Specific Variable • 2.4 Solve Mixture and Uniform Motion Applications • 2.5 Solve Linear Inequalities • 2.6 Solve Compound Inequalities • 2.7 Solve Absolute Value Inequalities • 4 Systems of Linear Equations • Introduction • 4.1 Solve Systems of Linear Equations with Two Variables • 4.2 Solve Applications with Systems of Equations • 4.3 Solve Mixture Applications with Systems of Equations • 4.4 Solve Systems of Equations with Three Variables • 4.5 Solve Systems of Equations Using Matrices • 4.6 Solve Systems of Equations Using Determinants • 4.7 Graphing Systems of Linear Inequalities • 7 Rational Expressions and Functions • Introduction • 7.1 Multiply and Divide Rational Expressions • 7.2 Add and Subtract Rational Expressions • 7.3 Simplify Complex Rational Expressions • 7.4 Solve Rational Equations • 7.5 Solve Applications with Rational Equations • 7.6 Solve Rational Inequalities • 8 Roots and Radicals • Introduction • 8.1 Simplify Expressions with Roots • 8.2 Simplify Radical Expressions • 8.3 Simplify Rational Exponents • 8.4 Add, Subtract, and Multiply Radical Expressions • 8.5 Divide Radical Expressions • 8.6 Solve Radical Equations • 8.7 Use Radicals in Functions • 8.8 Use the Complex Number System • 9 Quadratic Equations and Functions • Introduction • 9.1 Solve Quadratic Equations Using the Square Root Pro...

Learning Outcomes • Explain the meaning and significance of Taylor’s theorem with remainder • Estimate the remainder for a Taylor series approximation of a given function Taylor’s Theorem with Remainder Recall that the [latex]n[/latex]th Taylor polynomial for a function [latex]f[/latex] at [latex]a[/latex] is the [latex]n[/latex]th partial sum of the Taylor series for [latex]f[/latex] at [latex]a[/latex]. Therefore, to determine if the Taylor series converges, we need to determine whether the sequence of Taylor polynomials [latex]\left\\left(x\right)[/latex] as [latex][/latex] satisfies [latex]\begin\left(c\right)\left(x-a\right)[/latex]. Therefore, for some real number [latex]c[/latex] between [latex]a[/latex] and [latex]x[/latex]. It is important to note that the value [latex]c[/latex] in the numerator above is not the center [latex]a[/latex], but rather an unknown value [latex]c[/latex] between [latex]a[/latex] and [latex]x[/latex]. This formula allows us to get a bound on the remainder [latex]\left(x\right)|[/latex] is bounded by some real number [latex]M[/latex] on this interval [latex]I[/latex], then [latex]|\left(x\right)[/latex]. This theorem allows us to bound the error when using a Taylor polynomial to approximate a function value, and will be important in proving that a Taylor series for [latex]f[/latex] converges to [latex]f[/latex]. theorem: Taylor’s Theorem with Remainder Let [latex]f[/latex] be a function that can be differentiated [latex]n+1[/latex] times o...

• Math • Pure Maths • Remainder and Factor Theorems Remainder and Factor Theorems When dealing with polynomials of degree 3 or higher, it can often be quite difficult to factorize. Although this can be done through long division or synthetic division, it is always good to know a shortcut! In this section, we shall look at two new concepts called the Remainder Theorem and the Factor Theorem. We aim to apply these theorems to… Remainder and Factor Theorems • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ...

Remainder Theorem Questions Remainder theorem questions and solutions are provided here to help the students learn how to find the remainder when a polynomial is divided by another polynomial without performing division. Let’s practise questions on the Remainder theorem, and verify the answers with the solutions provided. Also, get additional practice problems on the Remainder theorem in this article. What is the Remainder theorem? Suppose p(x) is a polynomial of degree at least 1 and a is any real number. When p(x) is divided by (x − a), the remainder will be p(a). Thus, the Remainder theorem for polynomials can be written as: p(x) = (x – a) q(x) + r Here, p(x) = Dividend (x – a) = Divisor q(x) = Quotient r = Remainder Substituting x = a in p(x), we have; p(a) = (a – a) q(a) + r = 0.q(a) + r = r Hence proved. This can be compared with the basic division theorem, i.e., Dividend = Divisor × Quotient + Remainder. Also, read: Remainder Theorem Questions and Answers 1. Using the Remainder Theorem, find the remainder when x 6– 5x 4 + 3x 2 + 10 is divided by x – 2. Solution: Let p(x) = x 6– 5x 4 + 3x 2 + 10 Now, divide p(x) by x – 2. Here, remainder = r(x) = 6 Substitute x = 2 in p(x). p(2) = (2) 6– 5(2) 4 + 3(2) 2 + 10 = 64 – 80 + 12 + 10 = 86 – 80 = 6 Thus, r = p(a) = p(2) = 6 2. The polynomial 4x 2 – kx + 7 leaves a remainder of –2 when divided by x – 3. Find the value of k. Solution: Let p(x) = 4x 2– kx + 7 From the given, -2 is the remainder when p(x) is divided ...