Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Polynomials ↓


Remainder Theorem


The remainder theorem is a fundamental concept in algebra, particularly in the study of polynomials. It provides a straightforward way to determine the remainder of a polynomial when divided by a linear polynomial of the form (x - a). This theorem simplifies the division process and is particularly useful when dealing with complex polynomials. In this lesson, we will understand the remainder theorem in a broad sense, with ample examples to aid understanding.

Understanding polynomials

Before diving into the remainder theorem, it is necessary to understand what polynomials are. A polynomial is an expression consisting of variables and coefficients, which are combined using only addition, subtraction, multiplication, and non-negative integer exponents of the variables.

Examples of polynomials: 1. 2x^3 - 3x^2 + 5 2. x^2 + 4x + 4 3. 3x + 7

Division of polynomials

When we talk about dividing polynomials, we refer to the process of dividing one polynomial by another. The result is typically a quotient and a remainder. For example, when you divide f(x) = x^3 - 3x^2 + 4x - 5 by (x - 2), you perform polynomial long division. The remainder theorem simplifies part of this process.

Remainder theorem

The remainder theorem states that if a polynomial f(x) is divided by a linear divisor (x - a), then the remainder of this division is f(a). In other words, to find the remainder, simply substitute the value of a into the polynomial f(x).

Example: Consider the polynomial f(x) = 2x^3 + 3x^2 - 5x + 6. Use the Remainder Theorem to find the remainder when f(x) is divided by (x - 1). Solution: Substitute a = 1 into f(x): f(1) = 2(1)^3 + 3(1)^2 - 5(1) + 6 = 2(1) + 3(1) - 5(1) + 6 = 2 + 3 - 5 + 6 = 6 Thus, the remainder when f(x) is divided by (x - 1) is 6.

Illustration of the remainder theorem

Visual representations help to understand the theorem better. Consider polynomial division and remainder as shown below.

Quotient polynomial (linear divisor) (x – a) + remainder f(a)

This simple visualization shows the concept of polynomial division where for a given (x - a), the polynomial can be expressed as its quotient and remainder. The vertical line drives and divides the result into its elementary components.

Proof of the remainder theorem

Let's prove this theorem for a deeper understanding:

Dividing the polynomial f(x) by (x - a) gives:

f(x) = (x - a)q(x) + r

where q(x) is the quotient polynomial and r is the remainder. According to the polynomial division theorem, since the degree of the remainder cannot be greater than the divisor, for a linear divisor like (x - a), r must be a constant. Now:

f(a) = (a - a)q(a) + r = 0 * q(a) + r = r

Hence, it is proved that when f(x) is divided by (x - a) the remainder is simply the value of f(x) substituted at x = a, hence f(a).

Working example

Let us look at some more examples to further strengthen our understanding of the Remainder Theorem.

Example 1

Find the remainder when f(x) = 4x^4 - 2x^3 + x - 7 is divided by (x - 2).

Solution: f(2) = 4(2)^4 - 2(2)^3 + 2 - 7 = 4(16) - 2(8) + 2 - 7 = 64 - 16 + 2 - 7 = 43 Therefore, the remainder is 43.

Example 2

Find the remainder when f(x) = x^2 + 2x + 3 is divided by (x + 1).

Solution: Here, the divisor is (x + 1) which can be rewritten as (x - (-1)). So, a = -1 f(-1) = (-1)^2 + 2(-1) + 3 = 1 - 2 + 3 = 2 The remainder when f(x) = x^2 + 2x + 3 is divided by (x + 1) is 2.

Applications of remainder theorem

The remainder theorem is not limited to just finding remainders, but has many applications, including simplifying polynomial division, factoring polynomials, and finding roots. Here are some possible applications:

  • Checking if a number is a root: If f(a) = 0, then (x - a) is a factor of f(x).
  • In algebraic structures for solving higher polynomial equations.
  • It helps in synthetic division and makes calculations simpler.

Practice problems

To master the Remainder Theorem, let's test your skills with some practice problems:

  1. Find the remainder when f(x) = 3x^3 - x^2 + 2x + 1 is divided by (x - 3).
  2. What will be the value of remainder when f(x) = 7x^4 + 5x^3 - 3x + 9 is divided by (x + 2)?
  3. If f(x) = x^3 + 4x^2 + x - 6, find the remainder on dividing by (x - 1).

Conclusion

The Remainder Theorem is a powerful tool that simplifies the polynomial division process. Its application is widespread and affects a variety of polynomial operations and algebraic simplifications. By understanding and applying this theorem, students can solve complex polynomial expressions more efficiently and verify factorization steps more accurately.


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Remainder Theorem

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