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Unit 2: Polynomial and Rational Functions

Quadratics, higher-degree polynomials, their zeros and complex roots, and rational functions with their asymptotes, holes, and inequalities.

16 topics · 62 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Quadratic Functions and Vertex Form

  1. How it worksHow does vertex form a(x-h)2+k reveal a parabola's vertex without graphing anything?
  2. Worked exampleFind the vertex of f(x) = 2x2 - 8x + 3 by converting it to vertex form.
  3. Worked exampleFind the maximum height reached by a ball whose path is h(t) = -16t2 + 64t + 5.
  4. Common mistakeWhy isn't the vertex of a parabola just the point where it crosses the x-axis?
Your turn1 of 2
Write f(x) = 3x2 + 4x + 8 in vertex form.

Completing the Square

  1. How it worksHow does completing the square rewrite a quadratic as a perfect square plus a constant?
  2. Worked exampleFind the vertex form of f(x) = x2 + 6x + 5 by completing the square.
  3. Common mistakeWhy can't you complete the square by adding a number to only one side of an equation?
  4. Worked exampleFind the solutions to x2 - 4x - 1 = 0 by completing the square.
Your turn1 of 2
Write f(x) = x2 + 8x + 6 in vertex form (x-h)2 + k.

Power Functions and End Behavior

  1. How it worksHow does the sign of a power function's exponent and coefficient control which way its ends point?
  2. Worked exampleFind the end behavior of f(x) = -3x5 as x approaches positive and negative infinity.
  3. Common mistakeWhy doesn't a higher exponent always mean a steeper graph everywhere?
  4. Ties togetherHow does a power function's end behavior compare to the end behavior of a full polynomial sharing its leading term?
Your turn1 of 2
For f(x) = -3x5, what happens to f(x) as x approaches positive infinity, and why?

Polynomial Shape and Degree

  1. How it worksHow does the degree of a polynomial limit the number of turning points its graph can have?
  2. Worked exampleFind the maximum possible number of turning points for f(x) = x5 - 5x3 + 4x.
  3. Common mistakeWhy doesn't a degree-4 polynomial have to actually show all three of its possible turning points?
Your turn1 of 2
Find the maximum possible number of turning points for a degree-7 polynomial.

Zeros and Multiplicity

  1. How it worksHow does the multiplicity of a zero decide whether a graph crosses or touches the x-axis there?
  2. Worked exampleFind the zeros and their multiplicities for f(x) = (x-2)2(x+1)3.
  3. Common mistakeWhy doesn't every root of a polynomial cross the x-axis the same way?
  4. Common mistakeWhy doesn't a polynomial's total count of real zeros have to equal its degree?
Your turn1 of 2
For f(x) = (x-4)4(x+4)2, find the total number of zeros counted with multiplicity.

Dividing Polynomials by Long Division

  1. How it worksHow do you divide a polynomial by a binomial one term at a time using long division?
  2. Worked exampleFind the quotient and remainder when x3 - 2x2 - 5x + 6 is divided by x - 3.
  3. Worked exampleFind the quotient when x3 + 2x2 - x - 2 is divided by x + 1.
  4. Common mistakeWhy doesn't polynomial long division stop the moment the degrees of the two polynomials match?
Your turn1 of 2
Find the remainder when x3 + 4x2 + 6x + 8 is divided by x - 2.

Synthetic Division

  1. How it worksHow does synthetic division use the root and the coefficients to build the quotient row by row?
  2. Worked exampleFind the quotient of (2x3 - 3x2 - 11x + 6) divided by (x - 3) using synthetic division.
  3. Worked exampleFind the quotient of (x3 + 4x2 - 5) divided by (x + 5) using synthetic division.
  4. Common mistakeWhy doesn't synthetic division work directly when dividing by something like 2x - 3?
Your turn1 of 2
Use synthetic division to find the remainder when 3x3 + 4x2 + 8x + 3 is divided by x - r, as a function of r.

The Remainder and Factor Theorems

  1. How it worksHow does the Remainder Theorem connect f(a) to dividing a polynomial by x - a?
  2. Worked exampleFind f(2) for f(x) = x3 - 4x2 + x + 6 and use it to decide whether x - 2 is a factor.
  3. Common mistakeWhy doesn't a remainder of zero only tell you a number was a solution, rather than a whole factor?
  4. Ties togetherHow does the Factor Theorem confirm which candidates from the Rational Zeros Theorem are actually roots?
Your turn1 of 2
Find f(a) for f(x) = x3 - 4x2 + 6x + 8, as a function of a.

The Rational Zeros Theorem

  1. How it worksHow does the Rational Zeros Theorem build its candidate list from the factors of the constant and leading terms?
  2. Worked exampleFind all possible rational zeros of f(x) = 2x3 + 3x2 - 8x + 3.
  3. Common mistakeWhy doesn't the Rational Zeros Theorem guarantee that any of its candidates are actual roots?
Your turn1 of 2
For f(x) = 2x3 + 3x2 - 8x + 3, which of the following is NOT a valid candidate from the Rational Zeros Theorem?

The Fundamental Theorem of Algebra

  1. How it worksHow does the Fundamental Theorem of Algebra guarantee exactly n roots for a degree-n polynomial, counting multiplicity?
  2. Worked exampleFind how many roots, counted with multiplicity, a degree-5 polynomial must have.
  3. Common mistakeWhy doesn't the Fundamental Theorem of Algebra promise that every root is a real number?
  4. Ties togetherHow does the Fundamental Theorem of Algebra guarantee enough roots for the conjugate pairing rule to work?
Your turn1 of 2
Find how many roots, counted with multiplicity, a degree-7 polynomial must have.

Complex Zeros and Conjugate Pairs

  1. How it worksHow does the Complex Conjugate Root Theorem pair up non-real roots of a real-coefficient polynomial?
  2. Worked exampleFind all zeros of f(x) = x3 - 2x2 + 4x - 8 given that 2i is one of them.
  3. Common mistakeWhy doesn't a polynomial with real coefficients ever have just one non-real root all by itself?
Your turn1 of 2
f(x) = x3 - 4x2 + 6x - 4·6 has a real zero and a pair of non-real zeros satisfying x2 = -6. Find the real zero.

Rational Functions, Domain, and Holes

  1. How it worksHow does a common factor canceling from numerator and denominator create a hole instead of an asymptote?
  2. Worked exampleFind the domain and the hole in the graph of f(x) = (x2-9)/(x-3).
  3. Common mistakeWhy doesn't canceling the (x-3) factors erase the fact that x = 3 was never in the domain?
  4. Common mistakeWhy does the simplified graph still need an open circle at the hole even after the factors cancel cleanly?
Your turn1 of 2
Find the simplified form of f(x) = (x2 - 72)/(x - 7).

Vertical Asymptotes

  1. How it worksHow do you locate a vertical asymptote from the factors left in a rational function's denominator?
  2. Worked exampleFind the vertical asymptotes of f(x) = (x+1)/(x2-4).
  3. Worked exampleFind where f(x) = 1/(x2 - 16) has a vertical asymptote on each side of the origin.
  4. Common mistakeWhy isn't every zero of the denominator automatically a vertical asymptote?
Your turn1 of 2
Find the vertical asymptote on the positive side for f(x) = 1/(x2 - 72).

Horizontal and Slant Asymptotes

  1. How it worksHow does comparing the degrees of the numerator and denominator decide between a horizontal or slant asymptote?
  2. Worked exampleFind the slant asymptote of f(x) = (x2 - 3x + 2)/(x - 1).
  3. Worked exampleFind the horizontal asymptote of f(x) = (3x2 + 1)/(2x2 - 5).
  4. Common mistakeWhy can a graph actually cross its horizontal asymptote even though it can never cross a vertical one?
  5. Going furtherWhat happens to the end behavior when the numerator's degree is two or more higher than the denominator's?
Your turn1 of 2
Find the horizontal asymptote (y-value) of f(x) = (7x2 + 1)/(6x2 - 5).

Graphing Rational Functions

  1. How it worksHow do you combine intercepts, asymptotes, and holes into a full graph of a rational function?
  2. Worked exampleFind the intercepts and asymptotes and sketch the graph of f(x) = (2x-4)/(x+1).
  3. Going furtherWhat happens to a rational function's graph near a vertical asymptote where the sign of the factor flips?
  4. Common mistakeWhy doesn't a rational function's graph have to be symmetric just because its formula looks balanced?
Your turn1 of 2
Find the x-intercept of f(x) = (4x - 7)/(x + 8).

Polynomial and Rational Inequalities

  1. How it worksHow does a sign chart use zeros and undefined points to test each interval of a rational inequality?
  2. Worked exampleFind the solution set of the quadratic inequality (x-2)(x+3) < 0.
  3. Worked exampleFind the solution set of the rational inequality (x+1)/(x-2) ≥ 0.
  4. Common mistakeWhy can't you multiply both sides of a rational inequality by the denominator without checking its sign?
Your turn1 of 2
Find the smaller boundary value of the solution interval for (x - 7)(x + 6) < 0.