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AI HL · 12.4

Solving Equations with Technology

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12.4.4

Solving and forming polynomial equations

12.4.4

Solving and forming polynomial equations

Target: I can find the real roots of quadratic, cubic and quartic equations with the polynomial solver, including a repeated root and a missing power. (+4 more)

By the end of this section

Key points

Solving with the solver

  • Write the equation with 0 on the right-hand side. Enter the coefficients from the highest power down, with 0 for a missing power.
  • A degree n equation has up to n roots. A repeated root appears once or twice in the solver.
  • If every root contains i, the equation has no real roots. Write "no real roots".
  • When real and complex roots appear together, give only the real ones.

Forming the equation from a context

  • Name the unknown and write every length in terms of it.
  • Use the right rule: area or volume formula, Pythagoras, or a sequence formula.
  • Rearrange to polynomial form before you use the solver.
  • Reject roots that cannot happen: a negative length, a side that would be negative, or n that is not a positive whole number.
  • Put the answer back in the question as a check.

Sequences

  • Geometric sum: \(S_n=\frac{u_1(1-r^n)}{1-r}\). Multiply through by \(1-r\) to get a polynomial in r.
  • The polynomial may have the root r = 1. The formula does not work for r = 1, so reject it.
  • Arithmetic sum: \(S_n=\frac n2(2u_1+(n-1)d)\) gives a quadratic in n.

Fractions and negative powers

  • Multiply every term by the denominator, or by x for a power \(x^{-1}\).
  • The result must have whole-number powers only: it is then a polynomial equation.

On the calculator

Quadratics, cubics and quartics

Menu > Algebra > Polynomial Tools

  • Choose the degree of the equation.
  • Enter the coefficients from the highest power down, with 0 for a missing power.
  • Read all the roots the solver lists.

Screenshot and video to come

Real roots and no real roots

  • Roots that contain i are complex. Do not give them as answers.
  • If every root contains i, there are no real roots.
  • A graph that never meets the x-axis agrees.

Screenshot and video to come

Solving with a graph or the solver

  • For an equation such as a geometric sum, expand to polynomial form and use the solver.
  • Or graph each side and find where they meet, then compare.

Screenshot and video to come

Explore: roots of x³ − 2x² + a

Move the slider for a. The curve goes up and down. The roots are where it meets the x-axis. Watch them appear, join and disappear.

Real roots
Local maximum
Local minimum

Move the slider to move the curve.

  1. For which values of a does the curve cross the axis three times? What do the two turning points have to be doing?
  2. When the curve only touches the axis, what is special about that root? What does the solver show?
  3. Move a past the point where roots disappear. How many real roots are left? Where did the others go?
  4. Adding a to the equation moves the curve. Which way does a positive a move it, and why?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A triangle's height

The base of a triangle is 3 cm longer than its perpendicular height. The area is 35 cm². Find the height.

Working
  1. Let x be the height

    Height = x, base = x + 3

    Name the unknown and write the other side in terms of it.

  2. Area equation

    \(\frac12x(x+3)=35\)

    Area = ½ × base × height.

  3. Polynomial form

    \(x(x+3)=70\), so \(x^2+3x-70=0\)

    Multiply by 2, then move 70 across.

  4. Solve

    Coefficients 1, 3, −70: x = −10 or 7

    Reject the negative root. The height is 7 cm.

Video walkthrough coming
A geometric sum

A geometric sequence has first term 5. The sum of the first three terms is 48.75. Find the possible values of the common ratio r.

Working
  1. Sum formula

    \(S_3=\frac{5(1-r^3)}{1-r}=48.75\)

    Use the geometric sum formula with u₁ = 5 and n = 3.

  2. Clear the fraction

    \(48.75(1-r)=5(1-r^3)\)

    Multiply both sides by \(1-r\).

  3. Polynomial form

    \(5r^3-48.75r+43.75=0\)

    Expand, then collect everything on the left.

  4. Solve

    Coefficients 5, 0, −48.75, 43.75: r = −3.5, 1 or 2.5

    r = 1 came from multiplying by \(1-r\). It is not allowed, so r = 2.5 or r = −3.5.

Video walkthrough coming
Repeated root, no real roots

Solve (a) \(4x^2-20x+25=0\) and (b) \(x^2-2x+6=0\).

Working
  1. (a) Solve

    Coefficients 4, −20, 25: x = 2.5

    The solver gives one answer. The root is repeated: the graph touches the x-axis and turns.

  2. (b) Solve

    Coefficients 1, −2, 6: x = 1 ± 2.24i

    Both answers contain i. They are complex, so (b) has no real roots.

  3. (b) State

    No real roots

    The graph of \(y=x^2-2x+6\) stays above the x-axis.

Video walkthrough coming
An open box

A sheet of card is 45 cm long and 30 cm wide. A square of side x cm is cut from each corner and the sides are folded up to make an open box. The volume of the box is 2500 cm³. Find the possible values of x.

45 − 2xx30 − 2xxEFGHIJKLMNOP
Working
  1. Dimensions

    Length \(45-2x\), width \(30-2x\), height x

    Each side loses a square from both ends. The height is the cut-out size.

  2. Volume equation

    \(x(45-2x)(30-2x)=2500\)

    Volume = length × width × height.

  3. Polynomial form

    \(4x^3-150x^2+1350x-2500=0\)

    Expand the brackets, then subtract 2500.

  4. Solve

    Coefficients 4, −150, 1350, −2500: x = 2.5, 10 or 25

    Reject x = 25: the width would be negative. x must be between 0 and 15.

  5. State

    x = 2.5 or x = 10

    Both fit the card, so there are two possible boxes.

Video walkthrough coming
Pythagoras and a rejected root

A right-angled triangle has sides x cm, \((x+7)\) cm and \((2x-1)\) cm, with \((2x-1)\) cm the longest side. Show that \(x^2-9x-24=0\), then find x.

xx + 72x − 1ABC
Working
  1. Pythagoras

    \(x^2+(x+7)^2=(2x-1)^2\)

    The square on the longest side equals the sum of the other two.

  2. Expand

    \(2x^2+14x+49=4x^2-4x+1\)

    Expand both brackets carefully.

  3. Polynomial form

    \(0=2x^2-18x-48\), so \(x^2-9x-24=0\)

    Move everything to one side and divide by 2.

  4. Solve

    Coefficients 1, −9, −24: x = −2.15 or 11.2

    x = −2.15 gives a negative side. Reject it.

  5. State

    x = 11.2

    Check: the longest side is \(2x-1=21.3\), longer than both others.

Video walkthrough coming
A quartic with a missing power

Find the real roots of \(x^4-3x^3+5x=4\).

Working
  1. Right-hand side 0

    \(x^4-3x^3+0x^2+5x-4=0\)

    The x² term is missing, so its coefficient is 0.

  2. Solve

    Coefficients 1, −3, 0, 5, −4: x = −1.35, x = 2.43, x = 0.960 ± 0.543i

    A quartic has up to 4 roots. Two contain i.

  3. State

    x = −1.35 or 2.43

    Only the real roots are answers.

Video walkthrough coming