To match a factored polynomial to a graph, start with the zeros, the -values where the curve meets the -axis. A crossing calls for a single factor; a touch-and-turn calls for a squared factor. Then check whether the ends rise or fall: the right factors with the wrong sign flip the graph upside down.
Hints
- Hint 1
A zero is an -value where the graph has height . Find every place the curve meets the -axis, including the place where it touches the axis and turns back.
- Hint 2
A factor becomes at . A squared factor, , stays nonnegative on both sides of . Which zero is a touch rather than a crossing?
- Hint 3
For a fourth-degree polynomial, a positive leading coefficient, the number multiplying the highest power of , sends both ends up. Check the graph’s ends before choosing the sign in front.
Step-by-step
Match the zeros, touch, and end behavior
Step 1Read the zeros from the graph
No Desmos needed. Read where the curve meets the axis. A zero is an -value where . The curve meets the -axis at , , and , so its three zeros are , , and .
- Step 2
Turn the zeros into factors
A factor equals when . So each zero points to a factor:
- Step 3
Square the factor where the curve touches
The curve crosses at and , but touches and turns at . The factor stays nonnegative on both sides of , so it gives that touch. A squared factor makes a touch; a single factor makes a crossing. The product must have the form
- Step 4
Use the ends to choose the sign
Both ends of the graph rise, so its leading coefficient, the number multiplying the highest power of , is positive. This product’s highest-power term is , so the positive coefficient fits; would send both ends down. The equation that could represent the graph is
Choice A.