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Solve x⁴ - 16x³ + 95x² - 248x + 240 = 0

The solutions of x⁴ - 16x³ + 95x² - 248x + 240 = 0 are x = 3, 4, 4, 5 (repeated roots counted once each: 3, 4, 5).

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Graph of y = x⁴ - 16x³ + 95x² - 248x + 2401.522.533.544.555.566.551015202530x = 3x = 4x = 5y
Graph of y = x⁴ - 16x³ + 95x² - 248x + 240. The curve crosses the x-axis at x = 3, x = 4, x = 5, the roots found above.

How to work it out

  1. 1

    A quartic has up to 4 real roots. The rational root theorem says that any integer root of a polynomial with integer coefficients and leading coefficient 1 divides the constant term 240, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±16, ±20, ±24, ±30, ±40, ±48, ±60, ±80, ±120, ±240.

    x4−16x3+95x2−248x+240=0x^{4} - 16x^{3} + 95x^{2} - 248x + 240 = 0
  2. 2

    Test the candidates until one gives 0. x = 3 is a root. Divide by (x − 3) with synthetic division; the bottom row is the quotient and a 0 remainder.

    31−1695−24824003−39168−2401−1356−800\begin{array}{c|ccccc} 3 & 1 & -16 & 95 & -248 & 240 \\ & 0 & 3 & -39 & 168 & -240 \\ \hline & 1 & -13 & 56 & -80 & 0 \end{array}
  3. 3

    x = 4 is a root. Divide by (x − 4) with synthetic division; the bottom row is the quotient and a 0 remainder.

    41−1356−8004−36801−9200\begin{array}{c|cccc} 4 & 1 & -13 & 56 & -80 \\ & 0 & 4 & -36 & 80 \\ \hline & 1 & -9 & 20 & 0 \end{array}
  4. 4

    What is left is the quadratic x² - 9x + 20 = 0. Solve it (factoring or the quadratic formula):

    x2−9x+20=0⇒x=5, 4x^{2} - 9x + 20 = 0 \Rightarrow x = 5,\ 4
  5. 5

    So the polynomial factors as

    x4−16x3+95x2−248x+240=(x−3)(x−4)2(x−5)x^{4} - 16x^{3} + 95x^{2} - 248x + 240 = (x - 3)(x - 4)^{2}(x - 5)
LaTeX source of the formulas on this page
\[ x^{4} - 16x^{3} + 95x^{2} - 248x + 240 = (x - 3)(x - 4)^{2}(x - 5) = 0 \]
\[ x^{4} - 16x^{3} + 95x^{2} - 248x + 240 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -16 & 95 & -248 & 240 \\ & 0 & 3 & -39 & 168 & -240 \\ \hline & 1 & -13 & 56 & -80 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 4 & 1 & -13 & 56 & -80 \\ & 0 & 4 & -36 & 80 \\ \hline & 1 & -9 & 20 & 0 \end{array} \]
\[ x^{2} - 9x + 20 = 0 \Rightarrow x = 5,\ 4 \]
\[ x^{4} - 16x^{3} + 95x^{2} - 248x + 240 = (x - 3)(x - 4)^{2}(x - 5) \]

Candidates tested (computed)

Candidate xp(x)Root?
172no
-1600no
212no
-21260no
30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots1616
Product of the roots240240

Questions

What are the roots of x⁴ - 16x³ + 95x² - 248x + 240 = 0?

The solutions of x⁴ - 16x³ + 95x² - 248x + 240 = 0 are x = 3, 4, 4, 5 (repeated roots counted once each: 3, 4, 5).

How do you factor x⁴ - 16x³ + 95x² - 248x + 240?

(x − 3)(x − 4)(x − 5).

What is the sum of the roots?

16 (it equals −(-16) by Vieta's formulas).

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    Solve x⁴ - 16x³ + 95x² - 248x + 240 = 0: x = 3, 4, 5 | WordGPT