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The Factor Theorem. The Factor Theorem. Suppose that a polynomial is divided by an expression of the form ( x – a ) the remainder is 0. What can you conclude about ( x – a )?. If the remainder is 0 then ( x – a ) is a factor of the polynomial. This is the Factor Theorem :.
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The Factor Theorem Suppose that a polynomial is divided by an expression of the form (x – a) the remainder is 0. What can you conclude about (x – a)? If the remainder is 0 then (x – a) is a factor of the polynomial. This is the Factor Theorem: If (x– a) is a factor of a polynomial then substituting a in x will give an answer of zero. The converse is also true: If substituting a in for x gives an answer of zero then (x– a) is a factor of a polynomial.
The Factor Theorem And so 2x2 - 7x + 3 = (x - 3)(2x – 1) Use the Factor Theorem to show that (x - 3) is a factor of 2x2 - 7x + 3. (x - 3) is a factor of 2x2 - 7x + 3 if we substitute 3 in for x and get zero. 2x2 - 7x + 3 = 2(3)2 - 7(3) + 3 = 18 – 21 + 3 = 0 as required.
The Factor Theorem And so 3x2 + 5x – 2 = (x + 2)(3x – 1) Use the Factor Theorem to show that (x + 2) is a factor of 3x2 + 5x – 2. (x + 2) is a factor of 3x2 + 5x – 2 if we substitute –2 in for x and get zero. 3x2 + 5x – 2 = 3(–2)2 + 5(–2) – 2 = 12 – 10 – 2 = 0 as required.
Factoring polynomials The Factor Theorem can be used to factor polynomials by systematically looking for values of x that will make the polynomial equal to 0. For example: Factor the cubic polynomial x3 – 3x2 – 6x + 8. Let x3 – 3x2 – 6x + 8. Start by testing 1, then -1, then 2, etc. x3 – 3x2 – 6x + 8 = 1 – 3 – 6 + 8 = 0 (x – 1) is a factor of x3 – 3x2 – 6x + 8. Now long divide
