Polynomial Equations, Zeros, and Graphs

Objective: Read zeros from multiple factors.

Prerequisite skills: factoring, zero-product property, and graph coordinates. A polynomial written as a product is zero whenever one of its variable factors is zero. A repeated factor records multiplicity, but it does not create a different numerical root. Common error: counting repeated factors as different numerical zeros. Restrictions and interpretation: Count distinct roots separately from multiplicities; zeros of denominators are not polynomial roots.

Objective: Determine whether a graph crosses or touches at a root.

Near a simple real root, a factor changes sign and the graph crosses the axis. An even-multiplicity factor stays nonnegative on both sides, so the graph touches without crossing if the other factors are nonzero there. Common error: assuming every x-intercept must cross the axis. Restrictions and interpretation: Count distinct roots separately from multiplicities; zeros of denominators are not polynomial roots.

Objective: Use a polynomial value to constrain an unknown constant.

Dividing a polynomial P(x) by x−r leaves remainder P(r). In particular, x−r is a factor precisely when P(r)=0. Substitution gives a coefficient equation without long division. Common error: using P(−r) when the factor is x−r. Restrictions and interpretation: Count distinct roots separately from multiplicities; zeros of denominators are not polynomial roots. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.