1 Factorise polynomial expressions 2 3 2 Divide a polynomial by a linear or quadratic factor 2 3 3 Apply the remainder theorem 2 3 4 Apply the factor theorem
In this section, we will learn to use the remainder and factor theorems to factorise and to solve polynomials that are of degree higher than 2 Before doing so,
THE FACTOR THEOREM EXPLAINED Consider the function f(x) = x 2 – 6x + 8 If x = 0 then f(0) = 0 – 0 + 8 = 8 If x = 1 then f(1) = 1 2 – 6×1 + 8 = 3
Dividing Polynomials; Remainder and Factor Theorems In this section we will learn how to divide polynomials, an important tool needed in factoring them
4 1 Synthetic Division; the Remainder and Factor Theorems LEARNING OBJECTIVES The proof of this statement is based on a theorem that is the bedrock
Factor Theorem A polynomial f(?) has a factor ? 2 k if and only if f(k) 5 0 ? GUIDED PRACTICE for Examples 3 and 4 Divide using synthetic division
Let G be a multigraph on n vertices, possibly with loops An f-factor is a subgraph of G with degree f, at the ith vertex for i = 1 2,
Property 1 in Definition 3 4 establishes that i does act as a square at least one complex zero, z1, and as such, the Factor Theorem guarantees that f(x)
We can use Factor Theorem to factorize a cubic polynomial as explained below This is a convenient method particularly for factorization of a cubic polynomial
remainder and factor theorems to factorise and to solve polynomials that are of degree higher than 2 Before doing so, let us review the meaning of basic
and factor theorems to find factors of polynomials A26 This gives an easy way of finding the remainder when a polynomial is divided by (x – a) Examples 1
Example 1 Q Suppose f (x)=5x3 - 14x2 + 12x - 3 (i) Is (x - 2) a factor? (ii) Is (x - 1 ) a factor? 4 2 - Algebra - Solving Equations 4 2 8 - The Factor Theorem
Apply the Remainder and Factor Theorems EXAMPLE 1 Use polynomial long division When you divide a polynomial f(χ) by a divisor d(χ), you get a quotient
Example Lets factor the polynomial f(x)=4x4 − 8x3 − 3x2 + 7x − 2 • First we compile the list of all possible rational roots using the Rational Zero' Theorem