4. Exploring Algebraic Identities

9 topics2 h

Chapter 4 of NCERT Mathematics for Class 9: Exploring Algebraic Identities. Five sections — What an identity is, and how to see one, Identities as factorisation tools, Extending to three terms, Factorising quadratics and Building new identities, and using them. Nine videos, 2 h in all.

Worked answers to the 25 exercise questions in this chapter →

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Part 1

What an identity is, and how to see one

10 minAn identity holds for every value; an equation need notThree consecutive squares, add the outer two, subtract twice the middle: 2 every time. Three cases agreeing is not yet a reason.10 min · 12 partsहिंदी9 minReading (a + b)² off a partitioned squarePut one mark on a segment and build the square on it. Two cuts, four pieces — and the four pieces are the identity, already assembled.9 min · 12 partsहिंदी

Part 2

Identities as factorisation tools

10 minRecognising an expression as an identity in disguiseFactorising is not guessing until something works. Read the identity from right to left and it becomes a recogniser.10 min · 12 partsहिंदी

Part 3

Extending to three terms

10 min(a + b + c)² by substitution, and the square that proves itA new identity does not need a new proof if you can turn it into an old one in disguise. Do not start again — change a name.10 min · 12 partsहिंदी

Part 4

Factorising quadratics

10 minAlgebra tiles: factorising by rebuilding the rectangleFind two numbers that add to the middle term and multiply to the constant. Two rules arriving from nowhere — except they are one rectangle.10 min · 12 partsहिंदी11 minSplitting the middle term once the tiles come awayEveryone learns the two-number trick. Almost nobody is told where it comes from, or why it is allowed to make two demands at once.11 min · 12 partsहिंदी

Part 5

Building new identities, and using them

10 minCubes: (a ± b)³ from a cube cut into eight pieces1, 3, 3, 1 is not something to memorise. It is a count, and you can watch it being made by cutting one cube three times.10 min · 12 partsहिंदी11 minSum and difference of cubes, and the three-term cubic identityMultiply out and six products appear. Four of them die in two pairs, and that is not luck — the second bracket was built so they would.11 min · 12 partsहिंदी10 minSimplifying a rational expression, and the factor you must not cancelSimplifying a fraction with letters in it is not tidying, it is dividing. And there is exactly one number you may never divide by.10 min · 12 partsहिंदी

What you will be able to do

  • Carry out the chapter's opening trick on three sets of three consecutive square numbers and report that the outcome is 2 each time
  • Draw a segment of length a + b as a segment of length a followed by one of length b, and mark all three lengths
  • Given a three-term expression, propose values for a and b by inspecting the first and last terms
  • Obtain (a + b + c)² by replacing b + c with a single letter and applying the two-term identity
  • Compute the area of a rectangle whose sides are x + 3 and x + 4 both by distributivity and by counting tiles, and check the two agree
  • Compare coefficients of two quadratic expressions and extract the sum and product conditions on a and b
  • Multiply (a + b) by a² + 2ab + b² and collect the result into four terms
  • Multiply (x − y) by x² + xy + y² and account for every term that cancels
  • State what has to be true before a common factor may be cancelled

What this chapter assumes you already know

  • Multiplying two brackets by the distributive property, and collecting like terms afterwards
  • Solving a simple equation such as x² = 25 and knowing it has two solutions
  • Area of a rectangle as length times breadth, and of a square as side squared
  • That a length can be split into two parts and the parts named separately
  • Multiplying out a(b + c) by the distributive property
  • Squaring a two-digit number by ordinary multiplication, so the shortcut has something to beat
  • Taking a numerical common factor out of a sum of terms
  • Squaring a product such as 6x or 5p, and knowing (6x)² = 36x²

It builds directly on Why a debt times a debt is a fortune, What "rational" means, and why the denominator cannot be zero, Turning a situation into an expression: terms, variables, coefficients, Proof by contradiction: why √2 cannot be a ratio of integers and Univariate polynomials and what degree names.

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.

  • "It worked, so it is true."
  • "But a proof is just a very careful check."
  • "One counterexample is not enough to reject a rule."
  • "(a + b)² = a² + b²."
  • "(a + b)² is always the bigger one."
  • "The middle term is always added, because there is a plus in front."
  • "If the letters are lengths, the picture settles everything."
  • "The trick works because 1, 4, 9 are small."