Book A Free Math Class # Zeros of a Polynomial — Definition & Examples TL;DR The zeros of a polynomial are the values of xxx that make the polynomial equal to zero — the solutions of p(x)=0, also called its roots. This article defines them, shows how to find them, works six examples, and draws the sharp line between zeros of a polynomial and the zero polynomial. ## The Numbers That Make a Polynomial Vanish Every polynomial hides a small set of special inputs where its value collapses to exactly zero — and those inputs decide where its graph crosses the x-axis. Finding them is one of the oldest problems in algebra, tackled by al-Khwarizmi more than a thousand years ago. The **zeros of a polynomial** p(x) are the values of xxx for which p(x)=0. If p(a)=0, then aaa is a zero of the polynomial. These are exactly the same as the **roots** of the equation p(x)=0, and geometrically they are the **x-intercepts** — the points where the graph meets the x-axis. A **polynomial** is an expression like p(x)=x²−5x+6, built from a variable raised to whole-number powers with constant coefficients. Its zeros are the specific numbers you can substitute for xxx to make the whole expression evaluate to 0. ## Zeros of a Polynomial Versus the Zero Polynomial These two phrases sound almost identical and mean completely different things. Getting them mixed up is the single most common confusion on this topic, so pin the difference down first. | | Zeros of a polynomial | Zero polynomial | | --- | --- | --- | | **What it is** | The input values that make p(x)=0 | The polynomial that is 0 everywhere | | **Example** | For p(x)=x²−4, the zeros are x=2 and x=−2 | p(x)=0 (all coefficients are 0) | | **How many** | Finitely many (at most the degree) | It is a single specific polynomial | | **Degree** | Not applicable — zeros are numbers | Undefined (or taken as −∞) | | **Plain-English** | "Where does this polynomial hit zero?" | "The polynomial whose value is always zero" | The **zero polynomial** is the polynomial p(x)=0, where every coefficient is zero, so it outputs 0 for every input, and its degree is left undefined. By contrast, the **zeros of a polynomial** are the handful of xxx-values that a _non-zero_ polynomial sends to 0. One is a whole polynomial; the other is a set of numbers. ## How to Find the Zeros of a Polynomial Finding zeros always comes down to the same idea: set p(x)=0 and solve. The method depends on the degree. - **Linear polynomial** p(x)=ax+b: solve ax+b=0 directly, giving the single zero x=−b/a. - **Quadratic polynomial** p(x)=ax²+bx+c: factor into two brackets and set each to zero, or use the quadratic formula x=−b±√(b²−4ac)/2a when it does not factor neatly. - **Higher-degree polynomial**: pull out any common factor first, then factor by grouping or test likely roots; each linear factor (x−a) contributes one zero x=a. - **Checking a candidate**: substitute the value into p(x). If the result is 0, it is a zero (this is the verification used in Example 4). ## Properties of the Zeros of a Polynomial The zeros of a polynomial obey a few reliable rules that connect them to the polynomial's degree and coefficients: - **Number of zeros is capped by the degree.** A polynomial of degree n has at most n real zeros — a quadratic at most two, a cubic at most three. - **Every zero corresponds to a factor.** By the factor theorem, x=a is a zero exactly when (x−a) is a factor of p(x). - **Sum and product tie to the coefficients.** For a quadratic ax²+bx+c with zeros α and β: α+β=−b/a, αβ=c/a For a cubic ax³+bx²+cx+d with zeros α,β,γ, the sum is −b/a and the product is −d/a. - **Complex zeros come in pairs.** If a polynomial has real coefficients, any non-real zeros occur in conjugate pairs, so they are added or removed two at a time. ## Examples of the Zeros of a Polynomial ### Example 1 **Find the zeros of p(x)=x−5.** Set the polynomial equal to zero. x−5=0 ⇒ x=5. **The zero is x=5.** ### Example 2 **Find the zeros of p(x)=x²−9.** Factoring makes both roots visible. x²−9=(x−3)(x+3)=0 ⇒ x−3=0 ⇒ x=3; x+3=0 ⇒ x=−3. **The zeros are x=3 and x=−3.** ### Example 3 **Find the zeros of p(x)=x²−5x+6.** Factor into two brackets. x²−5x+6=(x−2)(x−3)=0 ⇒ x−2=0 ⇒ x=2; x−3=0 ⇒ x=3. **The zeros are x=2 and x=3.** ### Example 4 **Verify that x=4 is a zero of p(x)=x²−7x+12.** Substitute x=4. p(4)=4²−7(4)+12=0. Since p(4)=0, **x=4 is a zero.** ### Example 5 **Find the zeros of p(x)=2x²−8.** Take out the common factor first. 2x²−8=2(x²−4) = 2(x−2)(x+2)=0 ⇒ x=2 or x=−2. **The zeros are x=2 and x=−2.** ### Example 6 **A polynomial has zeros x=1 and x=−3. Write a polynomial with these zeros.** p(x)=(x−1)(x+3)=x²+2x−3. **One such polynomial is p(x)=x²+2x−3.** ## Why Zeros of a Polynomial Matter Zeros are the concept that makes a polynomial _solvable_ rather than merely writable. - **They locate the x-intercepts**, so the graph's shape follows from them. - **They connect to factors**: by the factor theorem, x=a is a zero exactly when (x−a) is a factor. That is why factorization and finding zeros are two views of the same task. - **They drive equation-solving**, which is the whole point of polynomial equations across physics, engineering, and economics. ## The Mistakes Students Make Most Often ### Mistake 1: Confusing zeros of a polynomial with the zero polynomial **Where it slips in:** A question asks for "the zeros of the polynomial" and the student describes the zero polynomial instead. **The correct way:** Zeros are the _numbers_ x=2,−2; the zero polynomial is a different object entirely. ### Mistake 2: Losing the negative root **Where it slips in:** Solving x²=k by taking a single square root. **The correct way:** Every even-power equation can have a negative solution too. Factoring makes both signs visible; the habit that fixes this is always factoring rather than square-rooting one side. ### Mistake 3: Treating a constant factor as a zero **Where it slips in:** Reading 2(x−2)(x+2)=0 and worrying about the 2. **The correct way:** A non-zero constant factor never produces a zero; only the variable factors do. ## Conclusion - The **zeros of a polynomial** are the xxx-values that make p(x)=0, the same as its roots and its x-intercepts. - They are numbers; the _zero polynomial_ is the polynomial 0 itself, and the two must not be confused. - Find zeros by setting p(x)=0 and factoring; each zero a corresponds to a factor (x−a). - A polynomial has at most as many zeros as its degree, and some have no real zeros at all.