Solve x³ + x² − 4x − 4 = 0
Polynomial equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −2, x = −1, x = 2 |
Step-by-step solution
9 steps-
1 Givenx^{3} + x^{2} - 4 x - 4 = 0
Solve for x.
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2 Rational root theorem: any rational root is ±(divisor of 4)/(divisor of 1)
If a polynomial with whole-number coefficients has a root p/q in lowest terms, then p divides the constant term and q divides the leading coefficient. That gives a short list of candidates to test, instead of guessing.
x \in \{ -1,\; 1,\; -2,\; 2,\; -4,\; 4 \} -
3 Test the candidates: x = -1 makes the polynomial 0, so (x + 1) is a factor
If substituting a number makes the polynomial 0, the factor theorem says that (x + 1) is a factor. Dividing it out leaves a polynomial one degree lower, which is easier to solve.
\left(-1\right)^{2} - 4 - 4 \left(-1\right) + \left(-1\right)^{3} = 0 -
4 Divide the polynomial by (x + 1) (synthetic division)
Dividing by the factor just found splits the polynomial into that factor times a polynomial one degree lower. Synthetic division is a shortcut for this long division that only works with the coefficients.
\left(x + 1\right) \left(x^{2} - 4\right) = 0 -
5 This is a quadratic in standard form ax² + bx + c = 0
Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = 0 and c = -4, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = 0,\quad c = -4 -
6 There is no x term, so isolate x²x^{2} = 4
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7 Take the square root of both sides — remember both signs
A number and its negative give the same result when raised to an even power: 3² = 9 and (−3)² = 9. So if something squared equals k, that something is √k or −√k. Forgetting the minus sign loses a solution.
x = \pm 2 -
8 Solutionsx = 2\quad \text{or} \quad x = -2
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9 Check\begin{aligned}x = -2:\quad 0 = 0\quad\checkmark\\ x = -1:\quad 0 = 0\quad\checkmark\\ x = 2:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −2 | 0 | 0 | ✓ |
| −1 | 0 | 0 | ✓ |
| 2 | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.