Standard form tells you what a polynomial does at the ends. Factored form tells you everything that happens in the middle. If you can read a factored polynomial, you can sketch its graph in under a minute without plotting a single table of values — and, going the other way, you can look at a graph and write down its equation. The whole trick rests on one fact you already know from quadratics: a product equals zero only when one of its factors equals zero.
Zeros from factored form
A zero of is an input that makes the output — in other words, an -intercept of the graph. If , then requires at least one bracket to be zero, so the zeros are , and . Watch the signs: the factor gives the zero , because you set and solve. A factor like gives — fractional zeros come from factors whose leading coefficient is not .
Multiplicity: how the curve meets the axis
The exponent on a factor is called the multiplicity of that zero, and it decides how the graph meets the -axis there. Think about why. Near , the factor is the only part changing sign; everything else is roughly a fixed number. If is odd, flips from negative to positive as you pass through, so the graph crosses. If is even, stays positive on both sides, so the graph comes down, touches the axis and turns back. Higher multiplicities also flatten the curve, because shrinks very fast when is close to .
| Multiplicity | Example factor | Behaviour at that zero | Sign change? |
|---|---|---|---|
| 1 | Cuts straight through the axis | Yes | |
| 2 | Touches the axis and turns back (a parabola-like bounce) | No | |
| 3 | Flattens out along the axis, then crosses | Yes | |
| 4 | Flattens out, then turns back | No |
Sketching from the factors
- Find the degree (add the multiplicities) and the leading coefficient (multiply the leading coefficients of the factors, including any number out front). Mark the end behaviour on both sides of your sketch first.
- Mark each zero on the -axis and label its multiplicity.
- Find the -intercept by evaluating — this anchors the vertical position and the size of the curve.
- Decide whether the curve is above or below the axis on each interval between zeros, remembering that only odd multiplicities flip the sign.
- Draw one smooth, unbroken curve through everything. Polynomials have no jumps, breaks or sharp corners.
The zeros alone do not pin down a polynomial. The functions and have exactly the same -intercepts, but the second is five times taller everywhere in between. The value of out front is a vertical stretch by a factor of , plus a reflection in the -axis if is negative. That is why a family of curves shares a set of zeros, and why one extra point is enough to identify which member you have.
Writing an equation from a graph
Reverse the process. Read each -intercept and judge its multiplicity from how the curve behaves there (cross, bounce, or flatten-and-cross). Write the skeleton , check that the multiplicities add up to a degree consistent with the end behaviour, then substitute any other known point — usually the -intercept — and solve for .