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
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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
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remainder and factor theorems to factorise and to solve Now consider another example of a cubic polynomial When 2 ? 3 +4 +5 was divided by
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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
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24 fév 2015 · Use long division to determine the other factors Page 6 6 February 24, 2015 Example Five Factor fully
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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
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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
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polynomial division Example 1 Divide 14 5 4 2 3 - - + x x x by 2 - x Start by writing the problem out in long division form 14 5 4
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Example 5 Applying the Remainder Theorem Let p(x) = ?2x 4 + 6x 3 + 3x ? 1 Use synthetic division to evaluate p(2) Solution From the remainder theorem
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The remainder theorem provides a short cut to this remainder without the use of The factor theorem enables us to quickly search for Example (5)
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