SM1.2 Polynomials
You can find the zeroes of a quadratic polynomial, read zeroes off a graph, and connect the zeroes to the coefficients in both directions.
Topics shown in module order.
A polynomial's zeroes are the inputs that make it zero, and on a graph they are exactly where the curve crosses the x axis. For a quadratic, the two zeroes and the three coefficients carry the same information in different clothes: the sum and product of the zeroes can be read from the coefficients without solving anything, and a quadratic can be rebuilt from its zeroes alone. That two-way connection is the whole chapter.
Work through these
Find the zeroes of several quadratics by factorising, and mark each pair on a sketch of the curve
Factorising and graphing are the same fact seen twice. Marking the zeroes on the sketch ties the algebra to the picture, so that later, when a question hands you only one of the two, you can supply the other.
For each quadratic, check the sum and product of its zeroes against the coefficient formulas
Checking the formulas on zeroes you have already found teaches you what they say while the numbers are still concrete. The formulas matter because many questions give the sum and product and never the zeroes themselves.
Build a quadratic from a given sum and product of zeroes, then verify it by solving
Running the connection backwards is where marks are usually earned. Verifying by solving closes the loop and catches the sign errors this construction invites.
From a drawn graph, state how many zeroes the polynomial has and roughly where they are
Counting crossings needs no algebra at all, which is the point: it is a check you can run on any answer in seconds. A quadratic that never crosses the axis has no real zeroes, and seeing that on the picture beats memorising the condition.
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