# How do you write a four term polynomial that can be factored by grouping?

e.g.:

We can construct such a polynomial by reversing the factorisation.

If we want clean factorisations like this, we could use the alternative pattern:

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Here is one way I've done it when writing exam questions.

Make sure that the first two terms have a common factor. Not a constant common factor, but something involving the variable.

Warning: This may lead to a polynomial that cannot be factored completely over the integers.

For example we might start with

We could use the following:

Neither of these factor can be factored over the integers.

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To write a four-term polynomial that can be factored by grouping, you can follow these steps:

- Write down the polynomial in the form ( ax^3 + bx^2 + cx + d ).
- Group the terms into pairs.
- Factor out the greatest common factor from each pair.
- Factor out the greatest common factor from the resulting binomials.
- Check if any further factoring is possible.

Example: ( 2x^3 + 4x^2 + 3x + 6 ) Grouping: ( (2x^3 + 4x^2) + (3x + 6) ) Factor: ( 2x^2(x + 2) + 3(x + 2) ) Common factor: ( (x + 2)(2x^2 + 3) ) This polynomial can be factored by grouping.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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