Polynomials Class 10: Notes and Important Questions

This guide walks you through Polynomials Class 10, chapter 2 of NCERT Class 10 Mathematics. You will learn what the degree and zeroes of a polynomial are, what zeroes look like on a graph, and how zeroes are linked to coefficients. Important questions with full answers and a list of common mistakes close the guide.

Polynomials and their degree

A polynomial is an algebraic expression made of variables and coefficients. In a polynomial p(x), the highest power of x is called the degree.

• 4x + 2 has degree 1.
• 2y² − 3y + 4 has degree 2.
• p(u) = 7u⁶ − 3u⁴ + 4u² + u − 8 has degree 6.

Expressions with a variable in the denominator or under a square root, such as 1/(x − 1) or √x + 2, are not polynomials.

Types by degree:

The word quadratic comes from "quadrate", meaning square. In each general form, a, b, c and d are real numbers.

Value and zeroes of a polynomial

If k is a real number, p(k) is the value you get by putting x = k in p(x).

Take p(x) = x² − 3x − 4. Then p(2) = 4 − 6 − 4 = −6.

A real number k is a zero of p(x) if p(k) = 0. For the same polynomial:

• p(−1) = 1 + 3 − 4 = 0
• p(4) = 16 − 12 − 4 = 0

So −1 and 4 are the zeroes of x² − 3x − 4.

For a linear polynomial ax + b (a ≠ 0), set ax + b = 0 to get x = −b/a. A linear polynomial has exactly one zero:

zero = −(constant term) / (coefficient of x)

Example: for 2x + 3, set 2x + 3 = 0, so x = −3/2.

What zeroes look like on a graph

• The graph of y = ax + b is a straight line. Its zero is the x-coordinate of the point where the line meets the x-axis. For y = 2x + 3, this point is (−3/2, 0).
• The graph of y = ax² + bx + c is a curve called a parabola. If a > 0 it opens upwards like a U. If a < 0 it opens downwards like an upside-down U.

The zeroes of a quadratic are the x-coordinates where the parabola meets the x-axis. There are three possibilities:

1. It cuts the x-axis at two distinct points: two zeroes.
2. It touches the x-axis at exactly one point: two equal (coincident) zeroes.
3. It does not meet the x-axis at all: no real zeroes.

To read zeroes from a graph: make a table of x and y values, plot the points, find where the graph crosses or touches the x-axis (where y = 0), and read off the x-coordinates.

Zeroes of cubic polynomials

A cubic polynomial can have at most three zeroes, so its graph meets the x-axis at most three times.

• y = x³ − 4x meets the x-axis at x = −2, 0 and 2. These are its three zeroes.
• y = x³ has only one zero, x = 0.

In general, a polynomial of degree n has at most n zeroes.

Zeroes and coefficients of a quadratic

If α and β are the zeroes of ax² + bx + c:

• Sum of zeroes: α + β = −b/a = −(coefficient of x) / (coefficient of x²)
• Product of zeroes: αβ = c/a = (constant term) / (coefficient of x²)

Example: x² + 7x + 10 = (x + 2)(x + 5), so the zeroes are −2 and −5. Here a = 1, b = 7, c = 10.

• Sum = −2 + (−5) = −7, and −b/a = −7/1 = −7. ✓
• Product = (−2)(−5) = 10, and c/a = 10/1 = 10. ✓

When you factorise by splitting the middle term, make sure the two split terms multiply to a × c.

Forming a quadratic from its zeroes

A quadratic polynomial with sum of zeroes S and product P is

kx² − Sx + P, where k is a non-zero constant.

With k = 1 the simplest form is x² − Sx + P.

Example: sum = −3, product = 2. Then x² − (−3)x + 2 = x² + 3x + 2.

Zeroes and coefficients of a cubic

If α, β, γ are the zeroes of ax³ + bx² + cx + d:

Check with x³ − 4x (a = 1, b = 0, c = −4, d = 0) and zeroes −2, 0, 2:

• Sum = −2 + 0 + 2 = 0, and −b/a = 0. ✓
• Pairs = (−2)(0) + (0)(2) + (2)(−2) = −4, and c/a = −4. ✓
• Product = 0, and −d/a = 0. ✓

Remember this

• Degree = highest power of the variable.
• Zero of p(x): a value k with p(k) = 0; on the graph, an x-coordinate where y = 0.
• Degree n means at most n zeroes.
• Quadratic: α + β = −b/a, αβ = c/a.
• Cubic: sum −b/a, pairs c/a, product −d/a.
• Quadratic from zeroes: kx² − Sx + P.

Important questions with answers

1. What is the degree of 7u⁶ − 3u⁴ + 4u² + u − 8? 6, because the highest power of u is 6.

2. Find the zero of 2x + 3. 2x + 3 = 0 gives x = −3/2.

3. Find the zeroes of x² − 2x − 8 and verify the relationship with the coefficients. Split the middle term: x² − 4x + 2x − 8 = x(x − 4) + 2(x − 4) = (x − 4)(x + 2). Zeroes: 4 and −2. Sum = 2 and −b/a = −(−2)/1 = 2. Product = −8 and c/a = −8/1 = −8. Verified.

4. Find the sum and product of the zeroes of 2x² − 8x + 6 without solving. a = 2, b = −8, c = 6. Sum = −(−8)/2 = 4. Product = 6/2 = 3.

5. Find a quadratic polynomial whose zeroes have sum 1/4 and product −1. x² − (1/4)x − 1. Multiplying by k = 4 gives 4x² − x − 4.

6. Find the product of the zeroes of 3x³ − 5x² − 11x − 3. a = 3, d = −3. Product = −d/a = −(−3)/3 = 1.

7. A polynomial of degree 4 can have at most how many zeroes? At most 4.

8. A parabola does not touch or cross the x-axis. How many real zeroes does the quadratic have? None, because there is no x-value where the polynomial equals zero.

9. True or false: every quadratic polynomial has two distinct real zeroes. False. It can have two distinct zeroes, two equal zeroes, or no real zeroes, depending on whether the graph cuts, touches or misses the x-axis.

Common mistakes to avoid

• Dropping the minus sign: the sum of zeroes is −b/a, not b/a. For a cubic, the product is −d/a.
• Reading the zero from the y-intercept. Zeroes are x-coordinates where y = 0.
• Forgetting that a ≠ 0. If a = 0, the degree changes.
• Calling √x + 2 or 1/(x − 1) a polynomial.
• Assuming every quadratic has two different zeroes.

Work through each example with a pencil and check every sign, then study this chapter with Joy to practise more.