Polynomial equations with the solver Problem set B Worked solutions

Solve \(x^3-3x^2=4x-2\).

• \(x^3-3x^2-4x+2=0\)
• Coefficients 1, −3, −4, 2
• Polynomial solver: x = −1.29, 0.397 or 3.90

Find the real roots of \(x^4-7x^2=3x-2\).

• \(x^4+0x^3-7x^2-3x+2=0\)
• Coefficients 1, 0, −7, −3, 2 (the x³ term is missing: enter 0)
• Polynomial solver: x = −2.31, −0.852, 0.364 or 2.80

Find the real root of \(3x^3+4x+10=0\).

• \(3x^3+0x^2+4x+10=0\)
• Coefficients 3, 0, 4, 10
• Polynomial solver: x = −1.20 and a complex pair
• Only x = −1.20 is real

Find the real root of \(x^3+2x^2=-4x-11\).

• \(x^3 + 2x^2 + 4x + 11=0\)
• Coefficients 1, 2, 4, 11
• Polynomial solver: x = −2.32 and a complex pair
• Only x = −2.32 is real

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

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

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

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

   • Polynomial solver, coefficients 3, −7, 2: x = 0.333 or 2

A fish tank is a cuboid. Its height is x dm, its width is \((x+2)\) dm and its length is \((x+5)\) dm. Its volume is 150 dm³.

1. (a) Show that \(x^3 + 7x^2 + 10x - 150=0\).

   • Volume = \(x(x+2)(x+5)\)
   • \(x(x+2)(x+5)=150\) expands to \(x^3 + 7x^2 + 10x=150\)
   • Subtract 150: \(x^3 + 7x^2 + 10x - 150=0\)

2. (b) Find the value of x.

   • Polynomial solver, coefficients 1, 7, 10, −150: x = 3.35 and a complex pair
   • Only the real root counts: x = 3.35

A cyclist rides 120 km at a steady speed of v km/h. A second cyclist rides the same 120 km at \((v+6)\) km/h and takes 1 hour less, so \(\frac{120}{v}-\frac{120}{v+6}=1\).

1. (a) Show that \(v^2+6v-720=0\).

   • Multiply every term by \(v(v+6)\): \(120(v+6)-120v=v(v+6)\)
   • \(720=v^2+6v\)
   • \(v^2+6v-720=0\)

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

   • Polynomial solver, coefficients 1, 6, −720: v = −30 or v = 24
   • A speed cannot be negative: v = 24 km/h

Given that \(3x-\frac{2}{x+2}=5\):

1. (a) show that \(3x^2 + x - 12=0\);

   • Multiply every term by \((x+2)\): \(3x(x+2)-2=5(x+2)\)
   • \(3x^2 + 6x - 2=5x+10\)
   • \(3x^2 + x - 12=0\)

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

   • Polynomial solver, coefficients 3, 1, −12: x = −2.17 or 1.84

The number of followers of a social media account, in thousands, is modelled by \(N=t^3-6t^2+15t\), where t is the time in months after the account opens. Find when the account first reaches 60 thousand followers.

• \(t^3-6t^2+15t=60\), so \(t^3 - 6t^2 + 15t - 60=0\)
• Coefficients 1, −6, 15, −60
• Polynomial solver: t = 5.30 and a complex pair
• Only the real root is a time: t = 5.30 months