2. Introduction to Linear Polynomials

9 topics1 h

Chapter 2 of NCERT Mathematics for Class 9: Introduction to Linear Polynomials. Three sections — Expressions, polynomials, degree, Linear patterns and linear models and Linear relationships and their graphs. Nine videos, 1 h in all.

Worked answers to the 39 exercise questions in this chapter →

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Part 1

Expressions, polynomials, degree

10 minTurning a situation into an expression: terms, variables, coefficientsA letter is not a number you have forgotten. It is a declaration that one quantity in the story is still free to move.10 min · 12 partsहिंदी9 minUnivariate polynomials and what degree namesRestrict yourself to one letter and a single number describes the whole expression. That number is the degree, and it predicts how the expression behaves.9 min · 12 partsहिंदी9 minWhat makes a polynomial linear, and the equation you get by fixing its valueFeed a linear polynomial evenly spaced inputs and the answers come out evenly spaced too — wherever you start. That regularity is what the name buys you.9 min · 12 partsहिंदी9 minA polynomial as an input–output machineEvery question so far has been what a polynomial is. Ask what it does and the rule comes apart from the values.9 min · 12 partsहिंदी

Part 2

Linear patterns and linear models

10 minA constant difference is the signature of a linear pattern1, 3, 5, 7, 9 — every gap is 2, and a constant gap is the fingerprint of a linear pattern. But knowing the gap is not yet knowing the rule.10 min · 12 partsहिंदी10 minLinear growth and linear decayA cost that climbs and a tank that empties look like opposites. They are the same rule with one character changed.10 min · 12 partsहिंदी

Part 3

Linear relationships and their graphs

10 minRecovering y = ax + b from two observationsTwo bills arrive and no price list came with them. Hidden behind them are a rate and a fixed fee, and two observations are exactly enough to recover both.10 min · 12 partsहिंदी10 minWhy two points are enough to draw the lineA rule has an answer for every input you can name, so drawing it cannot mean drawing all of it. Two rows of a table is the entire cost.10 min · 12 partsहिंदी10 minWhat a and b do to the line: slope, y-intercept, and parallel familiesThe two constants in y = ax + b do jobs that never interfere. The only way to prove that is to move one while pinning the other.10 min · 12 partsहिंदी

What you will be able to do

  • Build an expression from a described situation by naming the quantities that vary and writing the arithmetic that combines them
  • Decide whether a given expression uses one variable only, and so whether the chapter's definition applies to it
  • Recognise a linear polynomial from its degree and give examples in several letters
  • Evaluate a linear polynomial at a given input, including a negative input
  • Extend a drawn growing pattern by two or three further stages and tabulate the counts
  • Build a table of values for a given linear rule over a stated range of inputs
  • Explain why a relationship of the form y = ax + b has exactly two unknowns
  • Choose two convenient inputs for a linear equation and compute the matching outputs
  • Read a and b off an equation written in the form y = ax + b

What this chapter assumes you already know

  • Arithmetic with whole numbers and decimals, including multiplying a decimal by a whole number
  • Perimeter and area of a rectangle, and that a rectangle has two sides of each measurement
  • Area measured in square units, and that a per-square-metre cost is multiplied by an area
  • Substituting a number for a letter in a simple expression, from earlier classes
  • Multiplying a bracket by a term, as in x times (10 – x)
  • Index notation: what x³ means, and that x¹ is just x
  • The convention that any non-zero number raised to the power 0 equals 1
  • Reading a signed coefficient, so that the coefficient in –3x³ is –3

It builds directly on Two axes, an origin, and why the order of the pair matters and The four quadrants, and reading a point's signs off its position.

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.

  • "x is a specific unknown number I am supposed to find."
  • "4x means 4 and then x."
  • "The constant 3 can be absorbed into the other terms."
  • "Two letters means degree two."
  • "50lw is just another term like the others."
  • "The wire example gives one rectangle."
  • "Width is 20 – x."
  • "Letter-numbers and variables are two different things."