2.4.3
Remainder Theorem
Remainder Theorem
Remainder Theorem
Dividing polynomials can be a tricky process. The remainder theorem allows us to quickly find the remainder of a polynomial when it is divided by a binomial.
![Illustrative background for Division](https://image-v2.cdn.app.senecalearning.com/2020-05/ffa10b1c-e862-484f-908d-a94bd1d0568f/remainder,h_400,q_80,w_640.png)
![Illustrative background for Division ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-05/ffa10b1c-e862-484f-908d-a94bd1d0568f/remainder,h_400,q_80,w_640.png)
Division
Division
- A given polynomial can be decomposed into a quotient, divisor and remainder as follows:
- Where is the polynomial, is the quotient, is the divisor and is the remainder.
![Illustrative background for Division](https://image-v2.cdn.app.senecalearning.com/2020-01/f3756d44-1f59-43c8-b39e-e53e5b2b78e8/remainder-theorem-2,h_400,q_80,w_640.png)
![Illustrative background for Division ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/f3756d44-1f59-43c8-b39e-e53e5b2b78e8/remainder-theorem-2,h_400,q_80,w_640.png)
Division
Division
- When evaluated for the value , this decomposition can be written as:
- .
- This leads us to the remainder theorem.
![Illustrative background for Remainder theorem](https://image-v2.cdn.app.senecalearning.com/2020-01/5a36ee35-a808-4959-944a-6adb92a33eba/remainder-theorem-3,h_400,q_80,w_640.png)
![Illustrative background for Remainder theorem ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/5a36ee35-a808-4959-944a-6adb92a33eba/remainder-theorem-3,h_400,q_80,w_640.png)
Remainder theorem
Remainder theorem
- If the polynomial function is divided by , then the remainder is .
- We can use this to find the remainder of polynomials for given devisors without actually having to do the division!
- if and only if is a factor of .
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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