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C. Polynomial and Rational Functions · Polynomial Functions and Equations

Polynomial Functions and Equations: Unit Review

C1.1 C1.2 C1.3 C1.4 C1.5 C1.6 C1.7 C1.8 C1.9 C3.1 C3.2 C3.3 C3.4 C3.7 C4.1 C4.2 C4.3About 40 minutes
Learning goals
  • I can show what I know about polynomial functions, their graphs, division, factoring, equations and inequalities.
  • I can find which lessons I should revisit.

Words to know

polynomial function
f(x) = aₙxⁿ + ... + a₁x + a₀ with whole-number exponents and aₙ ≠ 0.
factor theorem
x − b is a factor of P(x) if and only if P(b) = 0.
order of a zero
How many times its factor repeats; odd order crosses, even order touches.

Let's review!

In this unit you met polynomial functions of any degree. You recognized them from equations and from finite differences, described their end behaviour, turning points, zeros and symmetry, told them apart from exponential and sinusoidal functions, sketched them from factored form, transformed y = x³ and y = x⁴, and wrote equations from families and conditions. Then you divided polynomials, used the remainder and factor theorems and special patterns to factor cubics and quartics, solved equations (exactly and by estimation), solved problems, and solved inequalities with graphs and sign charts.

The next unit, Rational Functions and Equations, divides one polynomial by another. Everything here, especially factoring and sign charts, is used again there. This review has a few questions from every lesson. Try each one, then press Show answer.

How to use this review

Do two or three sections at a time. Sketch a quick graph whenever you can: it catches most sign errors. Check factorizations by expanding or by substituting a number. For each question you miss, write down why (sign slip, wrong order of a zero, forgot to rearrange to 0, endpoint included by mistake) and go back to that lesson. Finish with the 10-question quiz.

Polynomial functionsdegree, leading coefficientfinite differencesGraphsend behaviour, turning pointszeros and their orderEquations of graphseven/odd, transformationsfamilies: y = k(x − r)...Divisionlong and syntheticP(x) = D(x)Q(x) + R(x)Remainder & factorremainder = P(b)P(b) = 0 ⟺ (x − b) factorEquations & inequalitiesroots = x-interceptssign charts, intervals
How the unit fits together: features and graphs of polynomials on top; the algebra tools (division, remainder and factor theorems) below, leading to equations and inequalities.

What Is a Polynomial Function?

1

Is f(x) = 3x⁻² + x a polynomial function?

2

Give the degree and leading coefficient of f(x) = (1 − 3x)²(x + 2)³.

3

For f(x) = x³ − 2x + 4, find f(−2).

Finite Differences and the Degree of a Polynomial

1

x = 0, 1, 2, 3, 4 and y = 2, 3, 10, 29, 66. Find the degree and leading coefficient.

2

What are the constant fourth differences of y = −3x⁴ + x (x-step 1)?

3

y = 4x² is tabulated with x-step 0.5. What are the constant second differences?

Key Features of Polynomial Graphs: End Behaviour, Turning Points and Zeros

1

Describe the end behaviour of f(x) = −x⁶ + 2x.

2

What numbers of turning points are possible for a quintic?

3

What kind of range does a quartic with a > 0 have?

Polynomial, Exponential or Sinusoidal? Telling Function Families Apart

1

x = 0, 1, 2, 3 and y = 5, 15, 45, 135. Which family?

2

x = 0 to 5 and y = 1, 4, 1, −2, 1, 4. Which family?

3

Which grows faster for large x, x² or 2ˣ?

Factored Form: Zeros, Their Order, and Sketching

1

Describe y = (x + 1)(x − 2)².

2

What is the sign of y = −(x − 1)(x + 3) for x > 1?

3

A cubic touches the x-axis at 0, crosses at 4 and passes through (2, −16). Find its equation.

Even and Odd Polynomial Functions: Symmetry

1

Is f(x) = x⁵ − 3x³ even, odd or neither?

2

Is f(x) = x⁴ − x + 1 even, odd or neither?

3

f is even and f(3) = 7. Find f(−3).

Transformations of y = x³ and y = x⁴

1

Give the vertex and range of y = 2(x − 1)⁴ − 5.

2

Where does (1, 1) on y = x³ go under y = −3(x + 2)³ + 4?

3

Find the x-intercept of y = (x − 3)³ + 8.

Families of Polynomial Functions and Equations from Given Conditions

1

Write the family of cubics with zeros −2, 1 and 5.

2

Find the member of that family with y-intercept 20.

3

A quartic has zeros ±√3, a zero of order 2 at x = 2, and y-intercept −12. Find its equation.

Dividing Polynomials: Long Division and Synthetic Division

1

Divide x³ − 5x² + 3x + 4 by x − 4.

2

Divide 2x³ + x − 7 by x + 2.

3

Divide x⁴ + x² + 1 by x² + x + 1.

The Remainder Theorem

1

Find the remainder when x⁴ − 2x + 3 is divided by x + 1.

2

Find the remainder when 4x² − 1 is divided by 2x − 3.

3

x³ + kx − 6 leaves a remainder of 4 when divided by x − 2. Find k.

The Factor Theorem: Factoring Cubics and Quartics

1

Factor x³ + 4x² + x − 6 fully.

2

List the possible rational zeros of 2x³ − x + 5.

3

Factor 2x³ − 3x² − 11x + 6 fully.

More Factoring: Sum and Difference of Cubes, Grouping and Quadratic Form

1

Factor 27x³ − 1.

2

Factor x⁴ − 10x² + 9.

3

Factor x³ + 3x² − 4x − 12.

Solving Polynomial Equations

1

Solve x³ − 6x² + 11x − 6 = 0.

2

Find all real roots of x³ + 27 = 0.

3

Solve x⁴ − 3x² − 4 = 0.

Solving Problems with Polynomial Models

1

A box has dimensions x, x + 1 and x + 3 (cm) and volume 72 cm³. Find x.

2

The product of three consecutive integers is 504. Find them.

3

An open box is made from a 16 cm by 16 cm sheet: V(x) = x(16 − 2x)². Find all x with V(x) = 288.

xy−2135−12−8−4412−124(0, 8)
Graph for the next review section: y = (x + 1)(x − 2)(x − 4), with zeros −1, 2, 4 and y-intercept 8.

Polynomial Inequalities: Reading Solutions from Graphs

1

Solve −3 < 1 − 2x ≤ 7.

2

Use the graph above to solve (x + 1)(x − 2)(x − 4) ≥ 0.

3

How do the solutions of (x + 1)(x − 2)(x − 4) = 0 and of > 0 differ?

Solving Factorable Polynomial Inequalities Algebraically

1

Solve x³ − 4x² − 5x < 0.

2

Solve (x + 2)²(3 − x) ≥ 0.

3

Solve x⁴ − 10x² + 9 ≥ 0.

Just for fun

You can now take a polynomial apart (factor it), find where it hits zero (solve it), and say where it is positive or negative (inequalities). That's a full check-up for any polynomial, no appointment needed.

Check your understanding

Quiz

1.

What is the degree of f(x) = (2x − 1)³(x² + 4)?

2.

The third differences of a polynomial (x-step 1) are all −12. What is its leading coefficient?

3.

Which describes the end behaviour of f(x) = −4x³ + x² − 7?

4.

Which function is even?

5.

What is the remainder when x³ + 2x² − 4 is divided by x − 2?

6.

Which is a factor of x³ − 3x² − 4x + 12?

7.

What is the factored form of x³ + 64?

8.

What are the real roots of x⁴ − 16 = 0?

9.

A cubic touches the x-axis at x = 3 and crosses at x = −1, with leading coefficient 2. Which equation fits?

10.

What is the solution of (x − 1)(x + 2)(x − 5) ≤ 0?

For parents and teachers

Use the review to find what to revisit:

  • Unsure what counts as a polynomial, or degree errors in products → Lesson 1 (What Is a Polynomial Function?).
  • Difference-table mistakes, or forgetting a·n!·hⁿ → Lesson 2 (Finite Differences).
  • Wrong end behaviour, or impossible numbers of turning points → Lesson 3 (Key Features of Polynomial Graphs).
  • Confusing x⁴ and 4ˣ, or wavy graphs called sinusoidal → Lesson 4 (Polynomial, Exponential or Sinusoidal?).
  • Sketches that cross at double zeros, or sign-chart errors → Lesson 5 (Factored Form: Zeros, Their Order, and Sketching).
  • Mixing up "even degree" and "even function" → Lesson 6 (Even and Odd Polynomial Functions).
  • Shifting the wrong way, or treating k = 2 as a stretch → Lesson 7 (Transformations of y = x³ and y = x⁴).
  • Errors finding k, or missing repeated factors → Lesson 8 (Families and Equations from Conditions).
  • Missing placeholders or sign errors in synthetic division → Lesson 9 (Dividing Polynomials).
  • Substituting x = 2 for the divisor x + 2 → Lesson 10 (The Remainder Theorem).
  • Not sure which values to test, or stopping before fully factored → Lesson 11 (The Factor Theorem).
  • Wrong signs in cubes patterns; missing grouping or quadratic form → Lesson 12 (More Factoring).
  • Losing a root by dividing by x, or reporting non-real roots as real → Lesson 13 (Solving Polynomial Equations).
  • Keeping roots outside the domain → Lesson 14 (Solving Problems with Polynomial Models).
  • Interval notation slips, or reversed signs in linear inequalities → Lesson 15 (Polynomial Inequalities: Reading Solutions from Graphs).
  • Sign charts with wrong signs, or endpoints included for strict inequalities → Lesson 16 (Solving Factorable Polynomial Inequalities Algebraically).

Looking ahead: the next unit, Rational Functions and Equations, builds on factoring and sign charts to study functions such as y = 1/(x − 2) and y = (2x + 1)/(x − 3), their asymptotes, and rational equations and inequalities. Polynomials return in Rates of Change and Combining Functions and Modelling, and in Calculus and Vectors (MCV4U), where you will differentiate them.