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IB Mathematics: Applications and Interpretation HL · 12.4: Solving Equations with Technology · 12.4.4

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Solving and forming polynomial equations Problem set B

Form the equation where needed, solve it with the polynomial solver, and give real roots to 3 significant figures. Reject any root that does not fit the context.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Use the polynomial solver to find the real roots of each equation.

    1. (a) \(x^2 + 7x + 5=0\)
    2. (b) \(36x^2 + 12x + 1=0\)
    3. (c) \(3x^2 - 4x + 7=0\)

  2. Find the real roots of each equation.

    1. (a) \(x^3 - 6x + 2=0\)
    2. (b) \(2x^3 + x^2 + 3x + 7=0\)

  3. Find the real roots of each equation.

    1. (a) \(x^4 - 8x^2 - 3x + 4=0\)
    2. (b) \(2x^4 - x^3 + 4x - 3=0\)

  4. Each equation can be written as a polynomial equation.

    1. (a) Show that \(2x+1=\frac{7}{x-3}\) can be written as \(2x^2 - 5x - 10=0\). Hence find the roots.
    2. (b) Show that \(x^2=5+3x^{-1}\) can be written in the form \(ax^3+bx^2+cx+d=0\), where a, b, c and d are whole numbers. Hence solve the equation.

In context

  1. The base of a triangle is 4 cm longer than its perpendicular height. The area is 30 cm². Find the height of the triangle.

  2. A geometric sequence has first term 10 and the sum of the first three terms is 15.6. Find the possible values of the common ratio r.

  3. A right-angled triangle has sides x cm, \((x+6)\) cm and \((2x-3)\) cm, with \((2x-3)\) cm the longest side.

    xx + 62x − 3ABC
    1. (a) Show that \(2x^2 - 24x - 27=0\).
    2. (b) Find x, and explain why one solution is rejected.

  4. A rectangular swimming pool measures 15 m by 6 m. A paved strip of width x m is laid outside the pool along one long side and one short side, as in the diagram. The area of the strip is 40 m².

    15 m6 mxxABCDEFG
    1. (a) Show that \(x^2+21x-40=0\).
    2. (b) Find x.

  5. A baker has a sheet of metal that measures 48 cm by 30 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open baking tray of volume 3000 cm³.

    48 − 2xx30 − 2xxEFGHIJKLMNOP
    1. (a) Show that \(4x^3 - 156x^2 + 1440x - 3000=0\).
    2. (b) Find the possible values of x.

  6. The volume of a cube, in cm³, and its surface area, in cm², are added together. The total is 400. Find the length of an edge.

Stretch

  1. A closed cylindrical gas canister has radius r cm and height h cm. Its volume is 250 cm³ and its total surface area is 240 cm².

    1. (a) Show that \(h=\frac{120-\pi r^2}{\pi r}\).
    2. (b) Hence show that \(\pi r^3-120r+250=0\).
    3. (c) Find the possible values of r.

  2. The first term of an arithmetic sequence is 5 and the common difference is 4. The sum of the first n terms is 945. Find n.

  3. The difference between two consecutive perfect cubes is 271. Find the possible pairs of cubes.

  4. n posts stand at the corners of a regular polygon. A rope is tied between every pair of posts that are not next to each other. The diagram shows the ropes from one post of a heptagon.

    ABCDEFG
    1. (a) Find an expression for the number of ropes.
    2. (b) There are 54 ropes. Find n.

jmaths.xyz · AI HL · 12.4.4 Solving and forming polynomial equations