Zeros
Find zeros of a polynomial function by setting f(x)=0 and solving: zeros are x-intercepts on the graph, and each corresponds to a factor (x-r) in the polynomial's factored form.
Key Concepts
The zeros of a quadratic function are the values of $x$ for which the function equals $0$. On a graph, the zeros are the $x$ intercepts, or the points where the graph intersects the $x$ axis.
For the function $f(x) = x^2 9$, the zeros are found by setting $x^2 9 = 0$, which gives $x = 3$ and $x = 3$. The function $f(x) = x^2 5x + 4$ has zeros at $x=1$ and $x=4$, because $(1)^2 5(1) + 4 = 0$ and $(4)^2 5(4) + 4 = 0$.
The 'zeros' are where the parabola party hits the ground! They are the special x values that make the function equal to zero, which means they are the exact points where the graph crosses the x axis. Finding the zeros helps you solve real world problems, like figuring out how long it takes for a launched rocket to land.
Common Questions
What are the zeros of a polynomial function?
The zeros of a polynomial function f(x) are the x-values where f(x)=0. On the graph, zeros appear as x-intercepts. In factored form, if f(x)=(x-r)(x-s), then x=r and x=s are zeros because substituting them makes f(x)=0.
How do you find the zeros of a polynomial algebraically?
Set f(x)=0 and solve. Factor completely using techniques like GCF, factoring trinomials, or synthetic division combined with the Rational Root Theorem. Apply the Zero Product Property: if a product equals zero, at least one factor must equal zero.
How do you use zeros to reconstruct the polynomial?
Each zero r corresponds to a factor (x-r). If the zeros of a polynomial are r1, r2, and r3, the polynomial is a*(x-r1)(x-r2)(x-r3) where a is the leading coefficient. Expand and simplify to write the polynomial in standard form.