Chapter 6: Polynomials

All sorts of polynomials and their rules explained.

Polynomial Function

A polynomial function is a function that can be written in the form

P(x) = a_nxⁿ + a_n-1x^n-1 + · · · + a₁x + a₀

Where n is a natural number or 0, and the coefficients a₀, . . . , a_n are real numbers, a_n≠0.

Parts of Polynomial Function

• Leading Term: Highest power of x.

• Degree: Power of the leading term.

• Monic: The leading term's coefficient is 1.

• Constant Term: Term without x.

• Zero Polynomial: P(x)=0, degree undefined.

Operations of Polynomials

• Addition: Add corresponding terms.

• Subtraction: Subtract corresponding terms.

• Multiplication: Distribute terms and combine like terms.

• Result: The sum, difference, and product of polynomials are always one single new polynomial.

Degree of Polynomial: Rules

The degree of a polynomial f, denoted as deg(f), follows these properties:

1. Sum of polynomials: deg(f + g) ≤ max(deg(f), deg(g))

2. Product of polynomials: deg(f × g) = deg(f) + deg(g)

Equating Coefficients

Two polynomials P and Q are equal only if their corresponding coefficients are equal. For two cubic polynomials, P(x) = a₃x³ + a₂x² + a₁x + a₀ and Q(x) = b₃x³ + b₂x² + b₁x + b₀, they are equal if and only if:

• a₃ = b₃

• a₂ = b₂

• a₁ = b₁

• a₀ = b₀

Division for Positive Integers

When p is divided by d, two integers, q the quotient and r the remainder, are obtained, such that

p = dq + r and 0 ≤ r < d.

E.g., dividing 27 by 4 gives 27 = 4 × 6 + 3.

In this example:

• 27 is the dividend;

• 6 is the quotient;

• 4 is the divisor;

• 3 is the remainder.

Note: If r = 0, then d is a factor of p. For example, 24 = 4 × 6.

Division of Polynomials

When dividing dividend P(x) by divisor D(x), quotient Q(x) and a remainder R(x) are obtained such that:

P(x) = D(x)Q(x) + R(x)

where R(x) has a degree less than D(x). If R(x) = 0, D(x) is a factor of P(x).

Equating Coefficients to divide

To divide a polynomial by equating coefficients:

1. Write the identity:

x³ - 7x² + 5x - 4 = (x - 3)(x² + bx + c) + r

= x³ + (b - 3)x² + (c - 3b)x - 3c

2. Equate the coefficients of each term (x², x, constant term) on both sides.

• For x²: b - 3 = -7 → b = -4

• For x: c - 3b = 5 → c = -7

• For the constant: -3c (-7) + r = -4 → r = -25

3. Plug in the value for b, c, and r.

Thus, the division is:

x³ - 7x² + 5x - 4 = (x - 3)(x² - 4x - 7) - 25.

Synthetic Division

1. Write the root of the divisor and the coefficients of the dividend.

2. Bring down the first coefficient.

3. Multiply the root by the number just brought down and add it to the next coefficient.

4. Repeat until you finish all coefficients. The last number is the remainder. Previous ones are coefficient of quotient (depends on degree).

Example: Divide x² + 3x + 2 by x + 1.

Set up:
Root = -1, Coefficients: [1, 3, 2]

Steps:

• Bring down 1.

• Multiply -1 × 1 = -1, then 3 + (-1) = 2.

• Multiply -1 × 2 = -2, then 2 + (-2) = 0.

Result: Quotient = x + 2, Remainder = 0.

*Synthetic Devision Table

-1 | 1 3 2

| -1 -2

----------------

1 2 0

Remainder Theorem

If P(x) is divided by βx + α, the remainder is P(-α/β).

Specifically, when P(x) is divided by x − α, the remainder is equal to P(α).

Factor Theorem

For a polynomial P(x):

• If P(α) = 0, then x − α is a factor of P(x).

• Conversely, if x − α is a factor of P(x), then P(α) = 0.
  More generally:

• If βx + α is a factor of P(x), then P(−α/β) = 0.

• Conversely, if P(−α/β) = 0, then βx + α is a factor of P(x).

Finding Factors of a Polynomial Using Integer Values

Assume a linear factor x - α. By considering the constant term, α must divide the constant. Test integer factors of the constant until P(α) = 0, which identifies α as a factor.

Rational-Root Theorem

Let P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + · · · + a₁x + a₀ be a polynomial of degree n with all the coefficients aᵢ integers. Let α and β be integers such that the HCF of α and β is 1 (i.e., α and β are relatively prime/coprime*).
If βx + α is a factor of P(x), then β divides aₙ and α divides a₀.

*Relatively Prime/Coprime

If α and β are relatively prime/coprime, they have no other common factors than 1. That is, their GCD (Greatest Common Divisor) is 1.

Difference / Sum of cubes

Difference of two cubes:
x³ - a³ = (x - a)(x² + ax + a²)

Sum of two cubes:
x³ + a³ = (x + a)(x² - ax + a²)

Transformations of f(x) = a(x − h)³ + k

For the graph of a cubic function of the form y = a(x − h)³ + k:

• Dilations: y = ax³; stretches if |a| > 1, compresses if 0 < |a| < 1.

• Reflections: Reflects if a < 0 (x/y-axis reflections are identical).

• Point of Inflection: At (0, 0) or (h, k) after shifts.

• Translations:

  • Vertical: y = x³ + k, up (+)/down (-) by k.

  • Horizontal: y = (x - h)³, right (-)/left (+) by h.

• Domain & Range: R.

• All cubic functions in the form of y = a(x − h)³ + k are one-to-one functions.

Inverse of f(x) = x³

• Inverse: f⁻¹(x) = x¹/³.

• Graphs: y = x³ and y = x¹/³ intersect at (1, 1) and (-1, -1).

• At x = 0: The graph of y = x¹/³ has a very steep slope, appearing nearly vertical.

How to Find an Inverse Function (Example)

• Given Function: f(x) = 2(x - 1)³ + 3.

• Step 1: Replace f(x) with y: y = 2(x - 1)³ + 3.

• Step 2: Swap x and y: x = 2(y - 1)³ + 3.

• Step 3: Solve for y:

  • Subtract 3: x - 3 = 2(y - 1)³.

  • Divide by 2: (x - 3)/2 = (y - 1)³.

  • Take the cube root: ∛((x - 3)/2) = y - 1.

  • Add 1: y = ∛((x - 3)/2) + 1.

• Result: The inverse is f⁻¹(x) = ∛((x - 3)/2) + 1.

One-to-One Functions

• A function is one-to-one if each input has a unique output and passes the horizontal line test (no horizontal line intersects the graph more than once).

• Only one-to-one functions have an inverse.

• Example: f(x) = x³ is one-to-one and has an inverse, f⁻¹(x) = x¹/³.

• Non-one-to-one: f(x) = x² fails the horizontal line test and has no inverse.

Solving Cubic Polynomials

Solve by factorisation techniques:

• Common factor: x³ - 3x² = x²(x - 3).

• Grouping: x³ + 3x² - 2x - 6 = (x³ + 3x²) - (2x + 6) = x²(x + 3) - 2(x + 3) = (x + 3)(x² - 2).

• Factor theorem: Test roots (e.g., f(1) = 0 implies x - 1 is a factor).

• Polynomial division: Divide to factor further.

• Sum/Difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²).

• Quadratic formula: Complete factorisation.

Graphs of Factorised Cubic Functions

• General form: y = ax^3 + bx^2 + cx + d.

• Cubic graphs are not simple transformations of y = x^3.

• Always have at least one x-axis intercept (Δ<0; 2 if Δ=0, 3 if Δ>0 for the quadratic after division).

• Y-axis intercept is at (0, d).

Steps to Sketch Factorised Cubic Functions

1. Find x-axis intercepts: Let y = 0 and solve for x.

2. Find y-axis intercept: Substitute x = 0.

3. Create a sign diagram: Determine the sign of y between and beyond the x-axis intercepts.

4. Analyze end behavior:

   • If the coefficient of x^3 (a) is positive, y increases as x → +∞ and decreases as x → −∞.

   • If a is negative, y decreases as x → +∞ and increases as x → −∞.

CAS: Sketching the graph

1. Open the Graph & Table application.

2. Enter the cubic equation (e.g., y = x^3 + 2x^2 − 5x − 6) into y1.

3. Tick the box to display the graph.

4. Adjust the window settings for better visibility (use Zoom Out/Zoom Box if needed).

5. Tap in the graph window and select Analysis > G-Solve > Max to find local maximum and Min for local minimum.

   • Ensure the turning points are visible on the graph before analysis.

Sign Diagram

A sign diagram is a number-line diagram that shows when an expression is positive or negative.

Use the sign diagram:

1. Identify intervals: Divide the number line based on x-axis intercepts.

2. Test signs: Substitute test points with any value from each interval into the cubic function (do NOT use x-int as they are places where the function is 0).

3. Mark signs: Record whether y is positive (+) or negative (−) for each interval.

4. Alternate signs: Signs switch around intercepts unless a repeated factor occurs (in that way, the sign stays ± depending on the factor).

Solving Cubic Inequalities

• Find roots using the factor theorem (e.g., x = 1).

• Factor the polynomial (e.g., P(x) = (x - 1)²(x + 3)).

• Identify intercepts (roots and y-axis intercept).

• See if there are any x-intercepts for the quadratic factor using the discriminant.

• Sketch the graph and analyse where y ≤ 0, <0, >0, or ≤0.

• Conclusion: Solve the inequality based on the graph (e.g., x ∈ (-∞, -3] ∪ {1}).

Families of Cubic Polynomial Functions

• y = ax³, where a > 0 (dilation of y = x³).

• y = a(x - h)³ + k, where a ≠ 0 (translation of y = ax³, inflection point at (h,k)).

• y = a(x - a)(x + b)(x - c), where a ≠ 0 (intercepts at x = a, -b, c).

• y = ax³ + bx² + cx, where a ≠ 0 (passes through the origin).

Finding rules for Cubic Equations

• Use f(x) = a(x - h)³ + k and one point of inflection by substitution.

• If x/y-intercepts and one other point are known (e.g., (x,y)), substitute x/y values into the equation.

• For f(x) = a(x - a)(x + b)(x - c), use known intercepts and a point; sub x/y values in.

• For graphs, use known points and intercepts to form an equation.

• Alternatively, if four points on the graph are known:

  • For f(x) = a(x - α)²(x - β), use α, β, and one point.

  • For f(x) = a(x - h)³ + k, use the inflection point (h, k) and one other point.

CAS: Solving for coefficients

• In the InM app, enter the expression ax³ + bx² + cx + d (use the Var keyboard for a, b, c, d).

• Highlight the expression and go to Interactive > Define.

• Tap the simultaneous equations icon on the Math1 keyboard, tap twice to expand for four equations.

• Enter the known values:

  • f(1) = 0

  • f(2) = -7

  • f(4) = 27

  • f(5) = 80

• Enter the variables a, b, c, d in the bottom right, separated by commas.

• Solve for the coefficients.

General Equation of a Quartic Function

A quartic function is of the form f(x) = a(x − h)⁴ + k.

• Translates the graph of y = x⁴:

  • h shifts it horizontally.

  • k shifts it vertically.

• The graph has a turning point at (h, k).

• If a < 0, the graph is inverted.

• The domain of all quartic functions is R (all real numbers).

• The range is not R for quartics unless a = 0.

  • If a > 0, the range is [k, ∞).

  • If a < 0, the range is (−∞, k].

Sketch the graph of Quartic Functions of the form y = a(x − h)⁴ + k

• Identify the turning point at (h, k).

• Find the x-axis intercepts by setting y = 0 and solving.

• Find the y-axis intercept by setting x = 0.

• Sketch the graph, considering that it may be inverted if a < 0.

E.g., y = (x − 2)⁴ − 1:

• Turning point: (2, -1).

• x-axis intercepts: Solve 0 = (x − 2)⁴ − 1:

  • x = 3 or x = 1.

• y-axis intercept: Set x = 0,

  • y = 16 − 1 = 15.

• Plot these points and sketch the graph.

Factorising and Solving Quartic Equations

Example: (using grouping method) x⁴ − 8x = 0

• Factor: x(x³ − 8) = 0.

• Solutions:

  • x = 0 or

  • x³ − 8 = 0 → x = 2.

• Final solutions: x = 0 or x = 2.

Even and Odd Functions

• Even polynomials: f(x) = f(−x), symmetric about the y-axis.

• Odd polynomials: f(x) = −f(−x), rotational symmetry around the origin.
  Examples:

  • Even: f(x) = x⁴.

  • Odd: f(x) = x³.

Even and Odd degree functions

Even-Degree Functions (e.g., f(x) = x², f(x) = x⁴):

• f(−x) = f(x)

• f(0) = 0

• As x → ±∞, y → ∞

• For m > n:

  • xᵐ > xⁿ for x > 1 or x < −1

  • xᵐ < xⁿ for −1 < x < 1

  • xᵐ = xⁿ for x = ±1 or x = 0

---

Odd-Degree Functions (e.g., f(x) = x³, f(x) = x⁵):

• f(−x) = −f(x)

• f(0) = 0

• As x → ∞, y → ∞, and as x → −∞, y → −∞

• For m > n:

  • xᵐ > xⁿ for x > 1 or −1 < x < 0

  • xᵐ < xⁿ for x < −1 or 0 < x < 1

  • xᵐ = xⁿ for x = ±1 or x = 0

Solving Quartic Inequality: x⁴ ≤ 2x³

1. Set up the inequality: x⁴ − 2x³ ≤ 0.

2. Factor: x³(x − 2) ≤ 0.

3. Solve for x:

   • x = 0 or x = 2.

4. The solution is x ∈ [0, 2].

Power Function

A power function is a function f with rule f(x) = x^r, where r is a non-zero real number. That is, r∈R∖{0}.

CAS 1: Finding values for x when specific value is provided

Example: V = x(12-2x)², V = 100

• Plot the graph of V = x(12 - 2x)² and V = 100 in the same window.

• Set graph window: xmin = 0, xmax = 6, and adjust using Zoom Auto.

• To find the intersection points:

  1. Go to Analysis > G-Solve > Intersection.

  2. Press the right arrow to find the other point of intersection.

CAS 2: Finding Maximum Values

Example: V = x(12-2x)²

• Use the graph of V = x(12 - 2x)² to find the maximum volume.

• Go to Analysis > G-Solve > Max.

• Ensure to remove the tick for y₂ and redraw the graph for the correct result.