Polynomial equations with the solver Problem set A Worked solutions

Solve \(x^3-2x^2=5x-3\).

• \(x^3-2x^2-5x+3=0\)
• Coefficients 1, −2, −5, 3
• Polynomial solver: x = −1.77, 0.520 or 3.25

Find the real roots of \(x^4-6x^2=2x-3\).

• \(x^4+0x^3-6x^2-2x+3=0\)
• Coefficients 1, 0, −6, −2, 3 (the x³ term is missing: enter 0)
• Polynomial solver: x = −2.09, −1, 0.572 or 2.51

Find the real root of \(2x^3+3x+8=0\).

• \(2x^3+0x^2+3x+8=0\)
• Coefficients 2, 0, 3, 8
• Polynomial solver: x = −1.28 and a complex pair
• Only x = −1.28 is real

Find the real root of \(x^3+x^2=-5x-9\).

• \(x^3 + x^2 + 5x + 9=0\)
• Coefficients 1, 1, 5, 9
• Polynomial solver: x = −1.54 and a complex pair
• Only x = −1.54 is real

Given that \(2x+\frac{3}{x}=7\):

1. (a) show that \(2x^2 - 7x + 3=0\);

   • Multiply every term by x: \(2x^2+3=7x\)
   • Move 7x across: \(2x^2 - 7x + 3=0\)

2. (b) hence find the possible values of x.

   • Polynomial solver, coefficients 2, −7, 3: x = 0.5 or 3

A block of wood is a cuboid measuring x cm by \((x+4)\) cm by \((x+7)\) cm. Its volume is 500 cm³.

1. (a) Show that \(x^3 + 11x^2 + 28x - 500=0\).

   • Volume = \(x(x+4)(x+7)\)
   • \(x(x+4)(x+7)=500\) expands to \(x^3 + 11x^2 + 28x=500\)
   • Subtract 500: \(x^3 + 11x^2 + 28x - 500=0\)

2. (b) Find the value of x.

   • Polynomial solver, coefficients 1, 11, 28, −500: x = 4.81 and a complex pair
   • Only the real root counts: x = 4.81

A train travels 240 km at a steady speed of v km/h. A faster train travels the same 240 km at \((v+20)\) km/h and takes 1 hour less, so \(\frac{240}{v}-\frac{240}{v+20}=1\).

1. (a) Show that \(v^2+20v-4800=0\).

   • Multiply every term by \(v(v+20)\): \(240(v+20)-240v=v(v+20)\)
   • \(4800=v^2+20v\)
   • \(v^2+20v-4800=0\)

2. (b) Find the speed of the slower train.

   • Polynomial solver, coefficients 1, 20, −4800: v = −80 or v = 60
   • A speed cannot be negative: v = 60 km/h

Given that \(2x-\frac{3}{x+1}=4\):

1. (a) show that \(2x^2 - 2x - 7=0\);

   • Multiply every term by \((x+1)\): \(2x(x+1)-3=4(x+1)\)
   • \(2x^2 + 2x - 3=4x+4\)
   • \(2x^2 - 2x - 7=0\)

2. (b) hence find the possible values of x.

   • Polynomial solver, coefficients 2, −2, −7: x = −1.44 or 2.44

The number of bacteria in a culture, in thousands, is modelled by \(N=t^3-6t^2+14t\), where t is the time in hours after the start. Find when the culture first reaches 40 thousand bacteria.

• \(t^3-6t^2+14t=40\), so \(t^3 - 6t^2 + 14t - 40=0\)
• Coefficients 1, −6, 14, −40
• Polynomial solver: t = 4.82 and a complex pair
• Only the real root is a time: t = 4.82 hours