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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 A

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.

Difficulty: first skills applying them stretch

Skills

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

    1. (a) \(x^2 - 8x + 3=0\)
    2. (b) \(25x^2 - 30x + 9=0\)
    3. (c) \(2x^2 - 3x + 5=0\)

  2. Find the real roots of each equation.

    1. (a) \(x^3 - 5x + 1=0\)
    2. (b) \(x^3 + 2x^2 + 3x + 5=0\)

  3. Find the real roots of each equation.

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

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

    1. (a) Show that \(3x-1=\frac{6}{x+2}\) can be written as \(3x^2 + 5x - 8=0\). Hence find the roots.
    2. (b) Show that \(x^2=4+2x^{-1}\) 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 5 cm longer than its perpendicular height. The area is 42 cm². Find the height of the triangle.

  2. A geometric sequence has first term 8 and the sum of the first three terms is 14. Find the possible values of the common ratio r.

  3. A right-angled triangle has sides x cm, \((x+4)\) cm and \((2x-2)\) cm, with \((2x-2)\) cm the longest side.

    xx + 42x − 2ABC
    1. (a) Show that \(2x^2 - 16x - 12=0\).
    2. (b) Find x, and explain why one solution is rejected.

  4. A rectangular lawn measures 12 m by 8 m. A path of width x m is laid outside the lawn along one long side and one short side, as in the diagram. The area of the path is 30 m².

    12 m8 mxxABCDEFG
    1. (a) Show that \(x^2+20x-30=0\).
    2. (b) Find x.

  5. A piece of card measures 36 cm by 24 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open box of volume 1500 cm³.

    36 − 2xx24 − 2xxEFGHIJKLMNOP
    1. (a) Show that \(4x^3 - 120x^2 + 864x - 1500=0\).
    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 300. Find the length of an edge.

Stretch

  1. A closed cylindrical tin has radius r cm and height h cm. Its volume is 300 cm³ and its total surface area is 300 cm².

    1. (a) Show that \(h=\frac{150-\pi r^2}{\pi r}\).
    2. (b) Hence show that \(\pi r^3-150r+300=0\).
    3. (c) Find the possible values of r.

  2. The first term of an arithmetic sequence is 4 and the common difference is 3. The sum of the first n terms is 924. Find n.

  3. The difference between two consecutive perfect cubes is 217. Find the possible pairs of cubes.

  4. A regular polygon has n sides. In the diagram, the dashed lines are the diagonals from one corner of a hexagon. A diagonal joins two corners that are not next to each other.

    ABCDEF
    1. (a) Find an expression for the number of diagonals of the polygon.
    2. (b) A polygon has 44 diagonals. Find n.

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