If f(x) is a polynomial and f(a) = 0, then (x–a) is a factor of f(x) Proof of the factor theorem Let's start with an example Consider 4 8 5

The Factor Theorem: Suppose p is a nonzero polynomial It is important to note that it works only for these kinds of divisors 5 Also Example 3 2 1

Now consider another example of a cubic polynomial divided by a linear divisor From the above example, we can deduce that: 2 ? 3 + 4 + 5 =

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

For example, the zeros of p are –3, 1, and 5, and the factors of p(x) are x + 3, x - 1, and x - 5 The following theorem generalizes this relationship

2 Remainder and Factor Theorems Interactive Mathematics Factor theorem state with proof examples and solutions factorise the Polynomials Maths Mutt

Example 8: 7 5 4 3)( 2 3 + ? + = x x x xf Find )4( ? f using (a) synthetic division (b) the Remainder Theorem Example 9: Solve the equation

4 2 - Algebra - Solving Equations 4 2 8 - The Factor Theorem Higher Level ONLY 1 / 5 Example 1 Q Suppose f (x)=5x3 - 14x2 + 12x - 3

24 fév 2015 · Use long division to determine the other factors Page 6 6 February 24, 2015 Example Five Factor fully

PROBLEM You Have the Right to the Remainder Theorem Chapter 5 Polynomial Expressions and Equations example are correct? - Long Division

linear factors corresponding to the zeros x=1,2 and 4 That is, Proof of the factor theorem Let's start with an example Consider 4 8 5 2 16 4 18 4 32 8 36

xf Example 8: 7 5 4 3)( 2 3 + − + = x x x xf Find )4( − f using (a) synthetic division (b) the Remainder Theorem Example 9: Solve the equation 06 11 3

The Remainder Theorem: Suppose p is a polynomial of degree at least 1 Now take the 2 from the divisor times the 6 to get 12, and add it to the -5 to get 7 The next example pulls together all of the concepts discussed in this section

polynomial division Example 1 Divide 14 5 4 2 3

p Use polynomial division in real-life problems 6 5 5 3 2 2 6 1 1 3 9 Your Notes REMAINDER THEOREM If a polynomial f(x) is divided by x

The Remainder Theorem and Factor Theorem The Factor Theorem A polynomial Example 5: Use f(x) = x3 – 7x2 + 15x – 9 to answer the questions below

Theorem 8 1: Let F be a field and non-zero be a non-constant polynomial in If and in, then Hence, in Example 5 for instance, we can write the remainder as

5 4 – Apply the Remainder and Factor Theorems Divide 247 / 3 8829 / 8 and Factor Theorems Example 1: Divide f(x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5

Extended version of the factor theorem: (ax + b) is a factor of a polynomial f(x) if Example: Show that (x – 3) is a factor of and find the other two factors

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