The student maps the algebraic zeroes of a polynomial to the x-coordinates of its graph's x-intercepts.
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Introduce the geometric definition of a zero.
Why are zeroes of a polynomial so important? To understand this, we need to look at their geometrical representations.
Geometrically, the zero of a polynomial is simply the -coordinate of the point where the graph of the polynomial intersects the -axis.
Graph showing a parabola intersecting the x-axis at two distinct points.
The zeroes of the quadratic polynomial are exactly the x-coordinates where the parabola crosses the x-axis.
Compare the 3 geometric cases for quadratic polynomials.
Worked example finding the number of zeroes from various graph shapes.
Problem. You are given three different graphs representing polynomials .
Find the number of zeroes for each polynomial based strictly on its graph.
Count intersections on a given curve to find the number of zeroes.
A polynomial graph is drawn on a coordinate plane. The curve intersects the -axis at exactly 2 distinct points and crosses the -axis at 1 point. How many zeroes does this polynomial have?
Guided problem mapping a table of values to x-intercepts to find zeroes.
Given the cubic polynomial , calculate its zeroes algebraically and explain how this maps to the x-intercepts on its graph.
State the polynomial given in the problem.
What condition must be met to find the algebraic zeroes?
Show your steps for substituting 0 and factoring the polynomial.
Solve the factored terms for x.
List the final zeroes.
How do these numerical answers map to the graphical intersections?