How are the remainder and factor theorems useful?

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How are the remainder and factor theorems useful?

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Answer 1 · 2014-08-14T20:36:11

You have only learned tools to factor quadratics; for higher degree polynomials, you can use long or synthetic division.

However, long/synthetic division is slow. If you guess the wrong divisor/factor, you will be wasting even more time.

For example: $P(x)=2x^2−x−1$ . (I am using a quadratic for the sake of brevity).

Using the remainder theorem, we can quickly get the remainder with $P(2)=2⋅2^2−2−1=8−2−1=5$ and $P(0)=2⋅0^2−0−1=−1$ . So, $x-2$ and $x-0$ are not factors of $P(x)$ . But consider that $P(0)$ is below the x-axis and $P(2)$ is above the x-axis; this means the $P(x)$ must cross the x-axis somewhere between 0 and 2.

This should lead you to try $P(1)$ as a factor. $P(1)=2⋅1^2−1−1=0$ , so $x−1$ is a factor of $P(x)$ by the factor theorem.

Once you have a factor, you can proceed with long/synthetic division to get your quotient. Then you will need to repeat and factor the quotient. Again, if it is a quadratic, switch to your quadratic factoring methods.

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How are the remainder and factor theorems useful?

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