Polynomials is the second chapter of Class 10 Maths, and it builds straight on Real Numbers. Once you understand what a “zero” really is and the neat link between the zeroes and the coefficients, most questions become quick. This page teaches the whole chapter in plain language, with examples, a plan, and an original practice set (with answers you can reveal) at the end.
What This Chapter Covers
- Zeroes of a polynomial and their geometrical meaning
- Relationship between zeroes and coefficients
- Forming a quadratic polynomial
- The division algorithm for polynomials
Your Game Plan for This Chapter
- First — Get clear on what a zero is and how it shows up on a graph.
- Next — Learn the sum and product relationship for a quadratic, and practise forming a quadratic from its zeroes.
- Last — Work through the division algorithm, then attempt the practice set below.
Study Notes
1. Zeroes of a Polynomial and Their Geometrical Meaning
A zero of a polynomial p(x) is a value k for which p(k) = 0. On a graph, the zeroes are exactly the x-coordinates of the points where the curve meets the x-axis. The degree of the polynomial sets the maximum number of zeroes it can have.
A polynomial can have at most as many zeroes as its degree — and each real zero is simply a point where its graph crosses the x-axis. A quadratic (a parabola) can cross the x-axis at 2 points, touch it at 1, or miss it entirely (0 real zeroes).
2. Relationship Between Zeroes and Coefficients
For a quadratic polynomial $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$, there is a lovely shortcut — you can find the sum and product of the zeroes straight from the coefficients, without solving anything.
For $ax^2 + bx + c$: Sum of zeroes $(\alpha + \beta) = -\frac{b}{a}$, and Product of zeroes $(\alpha\beta) = \frac{c}{a}$.
Example: for $x^2 – 5x + 6$, the zeroes are 2 and 3. Sum $= 5 = -\frac{-5}{1} = -\frac{b}{a}$. Product $= 6 = \frac{6}{1} = \frac{c}{a}$. Both check out.
Do not drop the minus sign. The sum of the zeroes is $-\frac{b}{a}$, not $\frac{b}{a}$. A missing minus is the single most common error in this chapter.
3. Forming a Quadratic Polynomial
Working backwards, if you know the sum (S) and product (P) of the zeroes, the quadratic is simply $x^2 – Sx + P$ (or any constant multiple of it).
Example: for zeroes 3 and $-2$, the sum is 1 and the product is $-6$, so the polynomial is $x^2 – (1)x + (-6) = x^2 – x – 6$.
“Find the zeroes and verify the relationship,” and “form a quadratic from a given sum and product,” are asked almost every year. Practise both until they are automatic.
4. The Division Algorithm for Polynomials
This topic has moved in and out of the rationalised CBSE syllabus in recent years, so please confirm whether it is in your current syllabus. We have kept it here so you have the complete reference in one place.
Just like dividing whole numbers (dividend = divisor × quotient + remainder), we can divide one polynomial by another: p(x) = g(x) × q(x) + r(x), where the remainder r(x) is either 0 or has a smaller degree than g(x). This is most useful when you already know one zero of a bigger polynomial and want to find the rest: divide out the known factor, and solve the simpler polynomial that remains.
Example: 2 is a zero of $x^3 – 3x^2 – 10x + 24$, so $(x – 2)$ is a factor. Dividing gives the quotient $x^2 – x – 12 = (x – 4)(x + 3)$. So the other two zeroes are 4 and $-3$.
Practice Worksheet
Try each question fully on your own first, then click Show Answer to check yourself.
Q1. Find the zeroes of $x^2 – 5x + 6$ and verify the relationship between the zeroes and the coefficients.
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Q2. Form a quadratic polynomial whose zeroes are 3 and $-2$.
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Q3. The sum of the zeroes of a quadratic is 4 and their product is 3. Find the polynomial and its zeroes.
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Q4. Find the zeroes of $6x^2 – 7x – 3$ and verify the sum and product.
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Q5. Given that 2 is a zero of $x^3 – 3x^2 – 10x + 24$, find the other zeroes.
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Once these feel easy, you have genuinely finished Polynomials. Do not aim for perfect on the first try — aim for one more correct question than yesterday.