Menu
Sign up Sign in

AI HL · 12.4

Solving Equations with Technology

0 of 45 skills done

0
12.4.3

Polynomial equations with the solver

12.4.3

Polynomial equations with the solver

Target: I can write a polynomial equation with 0 on one side and enter its coefficients in the polynomial solver, using 0 for a missing power. (+3 more)

By the end of this section

Key points

Polynomials and roots

  • A polynomial has only whole-number powers of x, from \(x^0\) upwards, such as \(3x^2-4x+1\).
  • The zeros of a polynomial are the roots of the equation "polynomial = 0".
  • If the highest power is \(x^n\), the equation has up to n roots.
  • Only real roots are answers. Roots that contain i (where \(i^2=-1\)) are complex: leave them out.

Using the polynomial solver

  • Write the equation with 0 on the right-hand side before you use the solver.
  • Enter the coefficients from the highest power down. A missing power has coefficient 0.
  • Take the coefficients from the rearranged equation: \(x^3+3x^2=7\) has constant term −7, not 7.
  • Give each root to 3 significant figures unless the question says otherwise.
  • A graph shows how many times the curve meets the x-axis. That is the number of real roots.

Rearranging to polynomial form

  • To remove a fraction, multiply both sides by the denominator. Multiply every term, including terms with no fraction.
  • In a "show that" question, write each line of rearranging. The result you are given is your check.
  • In a context, reject any root that cannot happen, such as a negative speed or length.

On the calculator

Solving a polynomial equation

Menu > Algebra > Polynomial Tools

  • Write the equation with 0 on the right-hand side.
  • Choose the degree, then enter the coefficients from the highest power down.
  • Type 0 for any missing power.
  • Read every root the solver lists.

Screenshot and video to come

Real roots among complex roots

  • The solver may list roots that contain i. These are complex.
  • Write down only the roots with no i.
  • Check against the graph: the real roots are where the curve meets the x-axis.

Screenshot and video to come

Checking with a graph

Menu > Trace > Graph Trace

  • Graph the polynomial and look for where it crosses the x-axis.
  • Count the crossings and compare with the real roots from the solver.

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.
Rearranging, then solving a cubic

Find the real roots of the equation \(x^3+3x^2=7\).

Working
  1. Right-hand side 0

    \(x^3+3x^2-7=0\)

    Move the 7 across. The solver needs 0 on one side.

  2. Every power

    \(1x^3+3x^2+0x-7=0\)

    There is no x term, so its coefficient is 0. Enter 1, 3, 0, −7.

  3. Solve

    Polynomial solver: x = 1.28

    A cubic has up to 3 roots. Here the other two roots are complex, so there is one real root. Give 3 significant figures.

Video walkthrough coming
Real root, complex roots ignored

Find the real root of the equation \(2x^3+x+6=0\).

−3−2−112−10−551015xyy = 2x³ + x + 6
Working
  1. Standard form

    \(2x^3+0x^2+1x+6=0\)

    The x² term is missing: enter 2, 0, 1, 6.

  2. Solve

    Polynomial solver: x = −1.33, x = 0.663 ± 1.35i

    The two answers with i are complex roots. Leave them out.

  3. Check with a graph

    Graph of \(y=2x^3+x+6\) crosses the x-axis once.

    One crossing means one real root. The sketch agrees with the solver.

  4. State

    x = −1.33

    Only the real root is the answer.

Video walkthrough coming
A fraction becomes a quadratic

Given that \(3x-\frac{4}{x}=5\): (a) show that \(3x^2-5x-4=0\); (b) hence find the positive value of x.

Working
  1. Multiply by x

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

    Multiply every term by x, including the 5 on the right. The fraction goes.

  2. Right-hand side 0

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

    This is the form to show. Move 5x across.

  3. Polynomial solver

    Coefficients 3, −5, −4: x = −0.591 or 2.26

    Part (b) uses the solver. Pick the positive root.

Video walkthrough coming