Polynomial Equations, Equality and Inequalities
A polynomial equation sets a polynomial equal to a value, usually 0, and solving it means finding all x that make the statement true; a polynomial inequality asks where the polynomial is positive or negative. In brief: to solve x3 − 4x = 0, factor to x(x − 2)(x + 2) = 0, so x = 0, 2 or −2. To solve x3 − 4x > 0, test the intervals between these zeros with a sign chart.
The method for equations is factoring plus the zero-product property. For inequalities, the zeros split the number line into intervals where the sign of the polynomial does not change. This guide also explains the equality of two polynomials, which means that all corresponding coefficients are equal, and a technique called comparing coefficients.
How do you solve polynomial equations?
- Move all terms to one side so that the other side is 0.
- Factor the polynomial.
- Use the zero-product property: if AB = 0, then A = 0 or B = 0.
- Solve each factor and check the solutions.
Example 1
x3 − 4x = 0 gives x(x2 − 4) = x(x − 2)(x + 2) = 0. The solutions are x = 0, x = 2 and x = −2.
Example 2: grouping
x3 − 3x2 − 4x + 12 = 0. Group: x2(x − 3) − 4(x − 3) = (x − 3)(x2 − 4) = (x − 3)(x − 2)(x + 2). The solutions are 3, 2 and −2.
Example 3: do not divide by x
x2 = 5x. Do not divide by x, which would lose x = 0. Write x2 − 5x = 0, so x(x − 5) = 0 and x = 0 or x = 5.
If the polynomial does not factor with integers, other tools are needed: for degree 2 the quadratic formula, and for higher degrees the rational root test and polynomial division.
When are two polynomials equal?
Two polynomials are equal as polynomials if all their corresponding coefficients are equal. This is a statement about the identity of expressions, valid for every x. It leads to the method of comparing coefficients.
Example 4
Find a, b, c so that a(x − 1)2 + b(x − 1) + c = 2x2 − 3x + 5. Expand the left side: ax2 − 2ax + a + bx − b + c = ax2 + (b − 2a)x + (a − b + c). Compare: a = 2; b − 2a = −3, so b = 1; a − b + c = 5, so c = 4. Check at x = 1: c = 4 and 2 − 3 + 5 = 4 ✓.
Another way to prove two polynomials of degree at most n are equal is to show they agree at n + 1 different values of x.
How do you solve polynomial inequalities?
- Move everything to one side: p(x) > 0 (or <, ≥, ≤).
- Find the zeros of p by factoring.
- Mark the zeros on a number line. They divide it into intervals.
- Check the sign of p in one test point of each interval.
- Choose the intervals with the required sign. Include the zeros if the inequality is ≥ or ≤.
Example 5: x3 − 4x > 0
The zeros are −2, 0 and 2. The intervals are (−∞, −2), (−2, 0), (0, 2) and (2, ∞). Evaluate x(x − 2)(x + 2) at x = −3: (−3)(−5)(−1) = −15, negative. At x = −1: (−1)(−3)(1) = 3, positive. At x = 1: (1)(−1)(3) = −3, negative. At x = 3: (3)(1)(5) = 15, positive. The solution is (−2, 0) ∪ (2, ∞).
Example 6: (x + 1)(x − 3)2 ≤ 0
The zeros are −1 and 3 (a double zero). The factor (x − 3)2 is never negative, so the sign is determined by x + 1, which is ≤ 0 for x ≤ −1. Also x = 3 makes the product 0. The solution is (−∞, −1] ∪ {3}.
Sign chart for x³ − 4x > 0
| Interval | x | x − 2 | x + 2 | Product |
|---|---|---|---|---|
| x < −2 | − | − | − | − |
| −2 < x < 0 | − | − | + | + |
| 0 < x < 2 | + | − | + | − |
| x > 2 | + | + | + | + |
The sign of the product changes at each zero of odd multiplicity and does not change at a zero of even multiplicity, such as the double zero in Example 6.
How do you solve equations by substitution?
Some higher-degree equations have a hidden quadratic form. For x4 − 5x2 + 4 = 0, let u = x2. Then u2 − 5u + 4 = 0, so (u − 1)(u − 4) = 0 and u = 1 or 4. Returning to x: x2 = 1 gives x = ±1 and x2 = 4 gives x = ±2. The four solutions are −2, −1, 1 and 2.
Where do these methods come from?
Solving polynomial equations is one of the oldest themes of algebra. Quadratic equations were solved in Babylon, cubic and quartic equations in sixteenth-century Italy. The sign chart for inequalities is a modern teaching technique that follows from the intermediate value property of continuous functions: a polynomial can change sign only at a zero.
Where are they used today?
Engineers solve polynomial inequalities to find ranges of parameters, such as the values of a design variable for which a profit polynomial is positive. Equations such as 4x3 − 24x2 + 32x = 0 appear in optimization of box volumes, and comparing coefficients is used in partial fractions and in matching formulas.
Which mistakes are common?
Not moving everything to one side
From (x − 1)(x − 2) = 6 you cannot conclude x − 1 = 6 or x − 2 = 6. Expand and set the polynomial equal to 0 first.
Losing a solution by dividing
Dividing by a variable factor loses the case where it equals zero. Factor instead.
Multiplying an inequality by a negative number
This reverses the inequality sign. Better to avoid multiplying by expressions with an unknown sign and use a sign chart.
Ignoring double zeros
Even multiplicity means the sign does not change, and the zero itself can satisfy a non-strict inequality.