Mathfolis

Equivalent Representations of Polynomial and Rational Expressions

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

Polynomials can be written in standard, factored, or vertex (for quadratics) form, and rational expressions can additionally be written as quotient + remainder/divisor. Each form makes a different structural feature obvious: standard form makes the leading term and y-intercept visible, factored form gives the zeros and their multiplicities, vertex form gives the vertex, and the quotient form (after long division) gives the slant asymptote.

Three forms of a quadratic

Standard
f(x)=ax2+bx+cf(x) = ax^2 + bx + c
Factored
f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2)
Vertex
f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k

Convert standard → vertex (complete the square)

For x2+bx+cx^2 + bx + c
x2+bx+c=(x+b2)2+c−b24x^2 + bx + c = \left(x + \tfrac{b}{2}\right)^2 + c - \tfrac{b^2}{4}
AP Tip: Match the form to the question. Asked for a vertex? Use vertex form. Asked for zeros? Use factored form. Asked for a slant asymptote? Long-divide.
Type 1

Convert standard form to factored form

Find two numbers whose product is the constant term and whose sum is the linear coefficient. Then write the quadratic as a product of two binomials.

Example 1
Factor f(x)=x2−5x+6f(x) = x^2 - 5x + 6. (A) (x−2)(x−3)(x - 2)(x - 3) (B) (x+2)(x+3)(x + 2)(x + 3) (C) (x−6)(x+1)(x - 6)(x + 1) (D) (x−1)(x−6)(x - 1)(x - 6)

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 2

Convert standard form to vertex form

Take half of the linear coefficient, square it, add and subtract it inside the expression, group the perfect square, and combine the constants.

Example 2
Rewrite f(x)=x2−6x+11f(x) = x^2 - 6x + 11 in vertex form. (A) (x−3)2+2(x - 3)^2 + 2 (B) (x+3)2−7(x + 3)^2 - 7 (C) (x−3)2−2(x - 3)^2 - 2 (D) (x−6)2+11(x - 6)^2 + 11

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 3

Choose the most useful form

Match the requested feature to the form that displays it directly. Avoid expanding only to re-extract the feature you started with.

Example 3
To find the maximum value of f(x)=−x2+4x+1f(x) = -x^2 + 4x + 1, which equivalent form is most directly useful, and what is the maximum value? (A) Standard form; max = 11 (B) Vertex form −(x−2)2+5-(x - 2)^2 + 5; max = 55 (C) Factored form; max = 55 (D) Vertex form −(x+2)2+5-(x + 2)^2 + 5; max = 55

Practice more of this type— AI-generated · always-new problems

Generate Problems →