Last month we looked at factorising cubics (and, by extension, higher order polynomials) by adapting the multiplication model of “farmers’ fields”. I ended by wondering what happens if we try to “factorise” using as our “factor”.
As before, we can express the given information as a field and see what happens:
The colour-coded diagram below shows the following steps:
Step 1 – to get the term, we just need ; this generates the term.
Step 2 – since we want a total quadratic term of , we have to have a term in our quadratic “factor”; this also generates the term.
Step 3 – since we want a total linear term of , the final entry in the top row must be
Thus our attempt to factorise hasn’t worked – we’ve ended up with a constant term of “” when a correct factorisation would have given us “”. However, we can instead deduce that:
To enable students to use the right language, I often use a numerical analogy, such as “what’s divided by ?” Since
the answer is sometimes given as “ remainder ”, or:
Your students will have known since primary school about remainders; they may not know that the “” in this calculation is called “the quotient”. Extending this language to our algebraic example, they can now say that “ divided by gives a quotient of and a remainder of ”.
There’s an important parallel here between numerical top-heavy fractions (if the numerator is greater than or equal to the denominator we can simplify the division, as we did with ) and algebraic ones (if the degree of the numerator is greater than or equal to the degree of the denominator we can simplify the division to find the quotient and remainder).
This seems like a good time for some practice: I’d ask students to find quotients and remainders using more fields. Just think how their algebraic fluency will improve as they do so! It’s well worth their considering a non-linear divisor: For example, Find the quotient and remainder when is divided by , leads to this field (where, as before, each column from left to right gets filled in, in turn, from top to bottom):
From which they conclude that
and so they say that the quotient is and that the remainder is .
Establishing the Remainder Theorem follows naturally. Recall that:
Since this is an identity, it is true for all values of . In particular, the students can notice that substituting into the original expression (the dividend, on the left-hand side) will have to give a value of , because the right-hand side will become – and they should check that it does.
Having by now seen lots of examples, your students should be confident in making the claim that if they attempt to divide some polynomial by a linear expression they will end up with , where (by experience) the degree of is one less than the degree of , and is constant. Thus they can substitute into both sides to see that by necessity. Do make sure that they learn to quote this result: the importance of “division by a field” is that it gives students conceptual understanding of division, but it’s not at all a procedurally fluent or efficient way to find the value of the constant remainder. The Factor Theorem, that is a factor of , is just a special case of the Remainder Theorem; again, students need to learn to quote this, not deduce it “from scratch” every time.
Division by quadratic divisors is usually a Further Maths topic, but it provides a good opportunity for single Maths students to deepen their confidence of the reasoning they’ve developed, and thus a question such as
When the polynomial is divided by the remainder is . When is divided by the remainder is . When is divided by the quotient is and the remainder is . Find and .
is well worth posing to them. Given all the examples they’ve considered, I’d expect that my students would be able to generalise from the linear case and produce a solution along the lines of
If you’re a frequent user of Geogebra, I’m sure you’ll have already envisaged the potential for presenting the “field routine”. Bernard Murphy (MEI Programme Leader) got in touch after last month’s article to share his polynomial-division file, which allows you to choose your coefficients and your divisor -
- and then runs through the process as you move the slider at the bottom of the screen:
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