Zeroes of a Polynomial Class 10: Meaning, Value of p(x) & How to Find

Quick answer: A zero of a polynomial p(x) is a real number k for which p(k) = 0 — that is, when you substitute it in, the polynomial’s value becomes zero. For example, 4 is a zero of p(x) = x² − 3x − 4 because p(4) = 16 − 12 − 4 = 0.

To get there you first need the value of a polynomial: p(k) is simply the number you get when you evaluate the polynomial by replacing x with k and simplifying. So for p(x) = x² − 3x − 4, p(2) = 4 − 6 − 4 = −6 and p(0) = −4. When that value comes out exactly 0, the number you substituted is a zero.

This guide covers the value p(k), the meaning of a zero, how to test whether a number is a zero, and how to find the zero of a linear polynomial — the essentials from Chapter 2 (Polynomials), Section 2.1.

Note on accuracy: This explanation follows the latest NCERT Class 10 Mathematics terminology for the value of a polynomial and its zeroes.

What is the value of a polynomial, p(k)?

If p(x) is a polynomial and k is any real number, then p(k) is the value of the polynomial at x = k — the result of substituting k for every x and simplifying. This step is called evaluating the polynomial.

Worked example. Let p(x) = x² − 3x − 4.

• p(2) = 2² − 3(2) − 4 = 4 − 6 − 4 = −6
• p(0) = 0² − 3(0) − 4 = −4
• p(−1) = (−1)² − 3(−1) − 4 = 1 + 3 − 4 = 0

So p(2) = −6, p(0) = −4, and p(−1) = 0. That last one is special — the value came out 0.

🎬 Watch (30-sec Short, opens in a new tab): Can you find p(2)? — try the substitution before the timer runs out.

Substituting cleanly is a small skill that saves marks. The Topic Practice on examfront lets you rattle through p(k) values with instant checking.

What is a zero of a polynomial? (Meaning)

A real number k is a zero of a polynomial p(x) if:

p(k) = 0

In words: a zero is a value of the variable that makes the whole polynomial equal to zero. From the example above, p(−1) = 0 and you can check p(4) = 16 − 12 − 4 = 0, so both −1 and 4 are zeroes of x² − 3x − 4.

• Meaning in plain words: “plug it in, get zero.”
• Variable defined: k is any real number you test; p(k) is the resulting value.
• Condition: it is a zero only when the value is exactly 0.

Value vs zero — don’t mix them up

You substitute k and get…	What it means
p(k) = 0	k is a zero of p(x)
p(k) ≠ 0 (any other number)	k is not a zero; you just found a value

A “zero” of a polynomial is the same idea as a root of the related equation p(x) = 0. Class 10 uses the word zero for the polynomial.

How to check if a number is a zero (step by step)

The one-second zero test

Substitute the number into p(x)
             ↓
       Simplify to a value
             ↓
     Is the value exactly 0?
       ↓ YES          ↓ NO
  It's a zero ✅   Not a zero ❌

1. Write down the polynomial p(x).
2. Substitute the number for x to get p(k).
3. Simplify. If p(k) = 0, the number is a zero; otherwise it is not.

Worked example. Is 2 a zero of p(x) = x² − x − 2? p(2) = 2² − 2 − 2 = 4 − 2 − 2 = 0. Since the value is 0, yes, 2 is a zero.

Worked example. Is 2 a zero of q(x) = x² − 5x + 6? q(2) = 4 − 10 + 6 = 0, so 2 is a zero of q(x) as well. (You can check q(3) = 9 − 15 + 6 = 0 too, so 3 is the other zero.)

Can you spot the zero?

For p(x) = x² + 3x + 2, which of these are zeroes?

①  x = −1
②  x = 0
③  x = −2
④  x = 1

Answers: ① and ③ are zeroes — p(−1) = 1 − 3 + 2 = 0 and p(−2) = 4 − 6 + 2 = 0. p(0) = 2 and p(1) = 6, so ② and ④ are not zeroes.

🎬 Watch (30-sec Short, opens in a new tab): Is −1 a zero? — watch the signs when you substitute a negative.

Practise a full set with instant feedback in examfront Topic Practice.

How to find the zero of a linear polynomial

A linear polynomial has the form ax + b with a ≠ 0. To find its zero, set it equal to 0 and solve:

The zero of ax + b is x = −b / a = −(constant term) ÷ (coefficient of x).

• Meaning in words: the one value that makes the linear polynomial zero.
• Variables defined: a = coefficient of x; b = constant term; condition a ≠ 0.
• Why one zero: a degree-1 polynomial has exactly one zero.

Worked example 1. Find the zero of 2x + 3. Set 2x + 3 = 0 → x = −3/2. Using the formula, −b/a = −3/2. ✔

Worked example 2. Find the zero of 3x − 9. Set 3x − 9 = 0 → x = 3. Using the formula, −b/a = −(−9)/3 = 3. ✔

So the zero of a linear polynomial is directly tied to its coefficients — a first hint that all polynomials have zeroes linked to their coefficients.

🎬 Watch (30-sec Short, opens in a new tab): Find the zero of x−7! — the fastest linear-zero example.

That deeper relationship (for quadratics and cubics) is built up step by step inside examfront.

examfront Insight The most common slip here isn’t the algebra — it’s the sign. Students write +b/a or flip it to −a/b. The zero of ax + b is −b/a: the constant on top, the coefficient of x on the bottom, with a minus in front. examfront’s Mistake Identification catches this pattern instantly.

How many zeroes can a polynomial have?

The number of zeroes is capped by the degree:

Degree 1 (linear)     →  exactly 1 zero
Degree 2 (quadratic)  →  at most 2 zeroes
Degree 3 (cubic)      →  at most 3 zeroes
         …
Degree n              →  at most n zeroes

In general, a polynomial of degree n has at most n zeroes. A quadratic, for instance, can have two zeroes, one repeated zero, or no real zero at all — the geometrical reason for this (where the graph meets the x-axis) is the next topic, Section 2.2.

🎬 Watch (30-sec Short, opens in a new tab): Cubic polynomial quiz! — how many zeroes can a cubic have?

You can explore the graph view and the zeroes–coefficients formulas in the full Polynomials track on examfront.

Why zeroes matter

Finding zeroes is one of the most reused skills in Class 10 algebra. It leads directly into:

• the relationship between zeroes and coefficients (sum and product of zeroes),
• factorising a polynomial (a zero k means (x − k) is a factor),
• solving quadratic equations (ax² + bx + c = 0),
• reading graphs (zeroes are where the curve meets the x-axis),
• and a large share of board exam questions on Chapter 2.

Getting comfortable with p(k) = 0 now makes all of those later topics much easier.

🎬 Watch (30-sec Short, opens in a new tab): Zero or factor? — the link between a zero and a factor (p(2)=0 ⇒ (x−2) is a factor).

Key Takeaways

• The value of a polynomial p(k) is found by substituting x = k and simplifying (evaluating).
• A number k is a zero of p(x) exactly when p(k) = 0.
• To test a zero, substitute and check whether the result is 0.
• The zero of a linear polynomial ax + b is −b/a, and it has exactly one zero.
• A polynomial of degree n has at most n zeroes (linear exactly 1, quadratic ≤ 2, cubic ≤ 3).

Quick Facts

• Chapter: 2 — Polynomials (CBSE Class 10 Maths).
• Value of a polynomial: p(k) = substitute x = k.
• Zero: k with p(k) = 0.
• Zero of ax + b: x = −b/a (a ≠ 0).
• Max zeroes: equal to the degree n.
• Zero vs root: same value; “zero” is used for a polynomial, “root” for the equation p(x) = 0.

Common Mistakes

1. Confusing p(k) with a zero. p(k) is just the value; the number is a zero only if that value is 0.
2. Sign slips when substituting negatives. (−1)² = 1, and −3(−1) = +3 — mind the signs.
3. Getting the zero-of-linear formula backwards. The zero of ax + b is −b/a, not −a/b.
4. Assuming every quadratic has two real zeroes. It has at most two; some have one or none.
5. Thinking 0 is always a zero. 0 is a zero only if the constant term is 0 (i.e. p(0) = 0).

FAQ

What is a zero of a polynomial? A real number k with p(k) = 0. Substituting k makes the polynomial equal to zero.

How do I find the value of p(k)? Replace x with k and simplify. For p(x) = x² − 3x − 4, p(2) = 4 − 6 − 4 = −6.

How do I find the zero of a linear polynomial? Set ax + b = 0 and solve; the zero is x = −b/a. For 2x + 3, the zero is −3/2.

How many zeroes does a polynomial have? At most as many as its degree — exactly 1 for linear, up to 2 for quadratic, up to 3 for cubic.

Can a linear polynomial have two zeroes? No. Being degree 1, ax + b has exactly one zero, −b/a.

Is a zero the same as a root? Yes. “Zero” describes the polynomial; “root” describes the equation p(x) = 0. The value is the same.

Is 0 a zero of every polynomial? No. 0 is a zero only when p(0) = 0, i.e. the constant term is 0.

Related Concepts

• Types of polynomials — know linear, quadratic and cubic before finding their zeroes. See Types of Polynomials Class 10.
• Polynomials Class 10 (full chapter guide) — the complete Chapter 2 overview.
• Relationship between zeroes and coefficients — the next layer (sum and product of zeroes).
• Quadratic equations — solving ax² + bx + c = 0 builds directly on finding zeroes.

Learn in this order

✓ Types of Polynomials
        ↓
✓ Zeroes of a Polynomial          (you are here)
        ↓
✓ Relationship Between Zeroes and Coefficients
        ↓
✓ Division Algorithm for Polynomials

Watch & revise (30-sec Shorts)

Quick recap reels — each opens in a new tab so you stay on this page:

• Can you find p(2)?
• Is −1 a zero?
• Find the zero of x−7!
• Cubic polynomial quiz!
• Zero or factor?

Next step: test yourself with the Polynomials Chapter Quiz on examfront. If a zero comes out wrong, Mistake Identification pinpoints the exact step — a substitution slip, a sign, or the formula — so you fix the cause, not just the answer, and Progress Tracking shows what to learn next.