For each polynomial function, identify its graph from choices A–F.
$\text{ƒ}\left(x\right)=(x-2)(x-5)$

Verified step by step guidance

1. Start by identifying the zeros of the polynomial function ƒ(x) = (x-2)(x-5). The zeros are the values of x that make the function equal to zero. Set each factor equal to zero: x - 2 = 0 and x - 5 = 0.
2. Solve each equation to find the zeros: x = 2 and x = 5. These are the x-intercepts of the graph, meaning the graph crosses the x-axis at these points.
3. Determine the degree and leading coefficient of the polynomial. Since ƒ(x) is the product of two linear factors, it is a quadratic polynomial of degree 2. When expanded, the leading term will be \(x^2\), which has a positive coefficient.
4. Use the leading coefficient and degree to determine the end behavior of the graph. Because the leading coefficient is positive and the degree is even, the graph opens upwards on both ends.
5. Combine this information: the graph crosses the x-axis at x=2 and x=5, opens upwards, and is a parabola. Use these characteristics to match the function to the correct graph among choices A–F.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Understanding the general shape and behavior of polynomial functions helps in predicting their graphs.

Roots or Zeros of a Polynomial

The roots or zeros of a polynomial are the values of x that make the function equal to zero. For ƒ(x) = (x-2)(x-5), the roots are x = 2 and x = 5, which correspond to the x-intercepts on the graph.

Graphing Polynomial Functions

Graphing involves plotting key points such as roots and determining the end behavior based on the leading term. For a quadratic like ƒ(x) = (x-2)(x-5), the graph is a parabola opening upwards, crossing the x-axis at the roots.