lesson
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Imagine trying to map the path of a rollercoaster without calculating every single foot of track. If you know where the track touches ground level and where the two ends fly off into the sky, you can sketch the entire ride in seconds.
With polynomials, those ground-level points are the zeros, and the far tails define the end behavior.
How do we pinpoint exactly where a polynomial touches the ground?
Finding the Zeros
A zero of a function is an input x-value that produces an output of zero, meaning f(x)=0. On a coordinate plane, every real zero is an x-intercept where the graph crosses or touches the horizontal axis.
When a polynomial is written in factored form, such as f(x)=(x+2)(x−1)(x−4), set each factor to zero to find the intercepts at x=−2, x=1, and x=4.
📊Interactive diagram
What happens when a factor is repeated more than once?
Multiplicity and Local Behavior
The power of a factor is its multiplicity, which controls the local behavior of the graph at that specific intercept.
When a zero has a multiplicity of 1, the curve crosses the x-axis linearly like a straight line. If a zero has an even multiplicity (m=2,4,…), the curve is tangent to the x-axis and 'bounces'—it touches the axis and turns around without crossing.
When a zero has an odd multiplicity greater than 1 (m=3,5,…), the curve flattens into an inflection point as it crosses the x-axis.
Once you know the behavior at each intercept, how do you determine which direction the far ends of the graph point?