⭐ POLYNOMIALS – DETAILED EXPLANATION (CLASS 10 NCERT)

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1️⃣ What is a Polynomial?

A polynomial is an algebraic expression made using:

  • Numbers (constants)
  • Letters like x, y (variables)
  • Whole number powers (0, 1, 2, 3…)
  • Addition, subtraction, multiplication

👉 Example of polynomials:

  • 2x+32x + 32x+3
  • x2+5x−7x^2 + 5x – 7x2+5x−7
  • 3x3+2x2−x+43x^3 + 2x^2 – x + 43×3+2×2−x+4

These are polynomials because all powers of x are whole numbers (0,1,2…).

❌ What is not a polynomial?

If power of the variable is:

  • Negative
  • Fraction
  • Decimal

It is not a polynomial.

Examples:

  • x−2x^{-2}x−2 → negative power (not allowed)
  • x\sqrt{x}x​ → x1/2^{1/2}1/2, fraction power (not allowed)
  • 4x3.54x^{3.5}4×3.5 → decimal power (not allowed)

2️⃣ Terms of a Polynomial

Example:
5x3−2x2+x−95x^3 – 2x^2 + x – 95×3−2×2+x−9

  • Terms: 5x³, -2x², x, -9
  • Coefficients: 5, –2, 1, –9
  • Constant term = –9

3️⃣ Degree of a Polynomial

The highest power of x in the polynomial.

Examples:

  • 3x4−x+93x^4 – x + 93×4−x+9 → degree 4
  • 5x2+2x5x^2 + 2x5×2+2x → degree 2
  • 777 → degree 0 (constant)

4️⃣ Types of Polynomials

Based on degree:

DegreeTypeExample
1Linear3x+53x + 53x+5
2Quadraticx2+4x+4x^2 + 4x + 4x2+4x+4
3Cubicx3−2x+1x^3 – 2x + 1x3−2x+1

5️⃣ Zeroes of a Polynomial (Very Important)

Zeroes = values of x for which the polynomial becomes 0.

⭐ Example 1:

p(x)=x−3p(x) = x – 3p(x)=x−3

Put p(x) = 0:

x − 3 = 0
x = 3

So, 3 is the zero.

⭐ Example 2:

p(x)=x2−9p(x) = x^2 – 9p(x)=x2−9

Put p(x) = 0:

x² – 9 = 0
x² = 9
x = 3 or −3

Zeroes: 3 and –3

6️⃣ Number of Zeroes

  • A linear polynomial has 1 zero.
  • A quadratic polynomial has 2 zeroes (max).
  • A cubic polynomial has 3 zeroes (max).

This is based on degree.

7️⃣ Geometrical Meaning of Zeroes (Graph meaning)

Graphs help to see zeroes easily.

  • Zeroes are the x-coordinates where the graph cuts the x-axis.

Example:

  • If graph touches x-axis at 2 points → 2 zeroes
  • If graph touches at one point → 1 zero
  • If graph does not touch x-axis → 0 zeroes (for quadratic)

8️⃣ Relationship Between Zeroes and Coefficients (Most Important!)

This is only for quadratic polynomials.

Ifp(x)=ax2+bx+cp(x) = ax^2 + bx + cp(x)=ax2+bx+c

and zeroes are α and β, then:

✔ Sum of zeroes

α+β=−ba\alpha + \beta = -\frac{b}{a}α+β=−ab​

✔ Product of zeroes

αβ=ca\alpha\beta = \frac{c}{a}αβ=ac​

⭐ Example:

Polynomial:x2−7x+12x^2 – 7x + 12x2−7x+12

Here:
a = 1, b = –7, c = 12

  • Sum = −−71=7-\frac{-7}{1} = 7−1−7​=7
  • Product = 121=12\frac{12}{1} = 12112​=12

Actual zeroes of this polynomial are 3 and 4:

  • 3 + 4 = 7 ✔
  • 3 × 4 = 12 ✔

So the formula works.

9️⃣ Finding Polynomial When Zeroes Are Given

If zeroes are α and β, polynomial is:k(x−α)(x−β)k(x – \alpha)(x – \beta)k(x−α)(x−β)

usually take k = 1 for simplicity.

Example:

Zeroes: 2 and 5
Polynomial:(x−2)(x−5)(x – 2)(x – 5)(x−2)(x−5)

Expand:x2−7x+10x^2 – 7x + 10x2−7x+10

🔟 Division Algorithm for Polynomials

Just like numbers:

Dividend = Divisor × Quotient + Remainder

Example:

Divide x2−1x^2 – 1x2−1 by x−1x – 1x−1.

You get:

  • Quotient = x+1x + 1x+1
  • Remainder = 0

So,x2−1=(x−1)(x+1)+0x^2 – 1 = (x – 1)(x + 1) + 0x2−1=(x−1)(x+1)+0

🎯 Important NCERT Results to Remember

  1. Degree 1 → 1 zero
  2. Degree 2 → 2 zeroes
  3. Sum of zeroes = –b/a
  4. Product of zeroes = c/a
  5. Polynomial in terms of zeroes: (x – α)(x – β)
  6. Zeroes = x-coordinates where graph touches x-axis
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