What’s covered / Class 10 Mathematics / Ch 2 2. Polynomials 7 topics 2 h
Chapter 2 of NCERT Mathematics for Class 10: Polynomials . Three sections — Degree, value and zero, Zeroes as crossings and Zeroes against coefficients. Seven videos, 2 h in all.
Worked answers to the 3 exercise questions in this chapter: Exercise 2.1 · Exercise 2.2
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Part 1
Degree, value and zero 14 min Degree as the label that separates linear, quadratic and cubic Six numbers come out of a polynomial nobody shows — minus 2, minus 1, 6, 25, 62, 123 — and three rounds of subtracting between them name its degree as 3, with not an x in sight. 14 min · 14 partsहिंदी 13 min Substituting a number into a polynomial, and what makes it a zero Evaluate x squared minus 3x minus 4 at minus 1, and both signs flip: minus one squared is plus one, minus three times minus one adds three. Miss either flip and nought becomes minus 8. 13 min · 13 partsहिंदी Part 2
Zeroes as crossings 13 min A zero is exactly a point where the curve meets the horizontal axis A zero of a polynomial and a crossing of the horizontal axis are not two facts. They are one sentence written twice, and the argument that they are the same is three lines long. What the picture buys is not a way of finding zeroes - the arithmetic already does that - but an answer to a question the arithmetic cannot reach: how many are there, and roughly where. 13 min · 14 partsहिंदी 13 min Which way a parabola opens, and the three ways it can meet that axis x squared minus x plus one never crosses the horizontal axis, and that is not a failed calculation. Completed, it becomes a square plus three quarters — never negative, so never zero. 13 min · 14 partsहिंदी 13 min Why degree puts a ceiling on how many zeroes there can be A cubic can bend twice, so it can leave the horizontal axis and come back to it more often than a parabola ever could. But not without limit. Every zero hands over a factor, each factor costs a degree, and a divisor cannot outrank what it divides - so the degree caps the count. That is a ceiling, not a promise: three cubics here carry three zeroes, one zero and two. 13 min · 15 parts Part 3
Zeroes against coefficients 14 min What the sum and product of two zeroes reveal about a, b and c Factor a quadratic, add its two zeroes, and the answer is already sitting in the coefficients you started with. It is not a coincidence: build a quadratic as k times two brackets, match the coefficients, and a turns out to BE k - which is why the sum is minus b over a and the product is c over a, ratios rather than statements about b and c. 14 min · 14 parts 15 min The three symmetric relations that hold for a cubic A quadratic has two relations because there are two ways to combine two zeroes: add them, or multiply them. A cubic has three, because three zeroes can be taken one at a time, two at a time, or all three at once. Nothing new is invented - one more way of combining what was already there. 15 min · 14 parts What you will be able to do Identify the degree of a polynomial in one variable by locating the largest power of that variable which actually appears Compute the output of a polynomial at a given input, showing the substitution before the arithmetic Explain what the curve y = p(x) records, point by point Predict from the sign of the leading number whether the curve of a quadratic opens upward or downward, and give the reason Build a table of values for a cubic and locate the inputs that return 0 Factorise a quadratic by splitting the middle term, and read its zeroes off the factors Verify that a stated number is a zero of a cubic by substitution, showing the working What this chapter assumes you already know Reading a power: knowing that in 5x³ the 3 counts how many times x is used as a factor Adding, subtracting and multiplying terms that carry a letter That numbers such as √2, √3 and −3/2 are real numbers and may sit as coefficients Class IX work on polynomials in one variable and the idea of degree, which §2.1 of this chapter explicitly picks up again rather than building afresh Arithmetic with negative numbers, in particular that a negative number squared is positive and cubed is negative Order of operations: powers before multiplication, multiplication before subtraction Solving a one-step linear equation such as 2k + 3 = 0 Plotting a point from a coordinate pair, and reading a point's coordinates back off a grid Where people usually slip up Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.
"Degree counts the terms." "The first term tells you the degree." "1/(x − 1) has degree −1, so it is a polynomial of negative degree." "√x + 2 is fine because there is no fraction." "a ≠ 0 is a technicality nobody checks." "Only x can be the variable." "Coefficients have to be whole numbers." "p(2) means p multiplied by 2." 
