Quick questions on The remainder and factor theorems: the remainder on division by (x - a) is P(a), the factor test P(a) = 0, the integer-zero test, finding unknown coefficients and fully factoring a polynomial

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the factor theorem?

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A remainder of zero is special: it means the divisor goes in exactly, so

x - a

is a factor of

P(x)

. Combine that with the remainder theorem, which says the remainder is

P(a)

, and you get a one-step test for factors.

What is the integer-zero test?

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The factor theorem gives a test for a given number, but it does not say which numbers to try. For a polynomial with integer coefficients there is a sharp restriction that cuts the search to a short list.

What is fully factoring a polynomial?

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Putting the three ideas together gives a dependable hand method for factoring a cubic or quartic that does not factor by inspection:

What is handling a non-monic linear factor?

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A factor like

2x - 1

is linear but not of the form

x - a

. The fix is to find the value of

x

that makes it zero. Setting

2x - 1 = 0

gives

x = \tfrac{1}{2}

, so

2x - 1

is a factor of

P(x)

exactly when

P\!\left(\tfrac{1}{2}\right) = 0

. More generally

bx - c

is a factor when

P\!\left(\tfrac{c}{b}\right) = 0

, because

\tfrac{c}{b}

is the zero of that linear expression.

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