Univariate polynomials and what degree names

What to assume they know

• Turning a situation into an expression: terms, variables, coefficients — terms, variables, coefficients and constants of an expression, and the difference between counting letters and reading powers
• 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
• Ordering whole numbers, in order to pick the largest power present

What they should be able to do

• Decide whether a given expression uses one variable only, and so whether the chapter's definition applies to it
• State the degree of a one-variable polynomial by locating its highest power
• Name a polynomial as constant, linear, quadratic or cubic from its degree
• Give the coefficient of a named power in a polynomial, carrying the sign
• Give the coefficient of a power that does not visibly appear, and justify the answer 0
• Identify the constant term of a polynomial, including when it is negative
• Explain why a non-zero constant is assigned degree 0 rather than no degree
• Write down polynomials to order when a degree is specified

Where it usually goes wrong

• "Degree counts the terms." 5y³ + y² + 2y – 1 has four terms and degree 3; x⁴ – 3x³ + 6x² – 2x + 7 has five terms and degree 4; 4x has one term and degree 1. Show all three together — the two counts move independently.
• "Degree is the biggest coefficient." In 5y³ + y² + 2y – 1 the largest coefficient is 5 and the degree is 3. Look up at the exponent, not along the line at the numbers.
• "A missing term has no coefficient." It has coefficient 0, which is why item 4 of Exercise Set 2.1 has an answer at all. Insert the 0z and the question stops being strange.
• "The coefficient of y² in 5y³ + y² + 2y – 1 is nothing." It is 1. An unwritten 1 is a convention, not an absence.
• "The coefficient of x³ in x⁴ – 3x³ + … is 3." It is –3. The sign belongs to the coefficient, not to the operation, and the chapter's own –1 constant makes the same point.
• "A plain number is not a polynomial." It is one, of degree 0. The chapter goes out of its way to fit 8 into the scheme rather than exclude it.
• "Zero itself must then have degree 0 too." The chapter says nothing about the polynomial 0, and neither should the explanation. 0 = 0x⁰ gives no largest power to read, which is why the case is usually handled separately — but that is beyond this chapter, and stating a rule for it here would be inventing one.
• "Changing the letter changes the polynomial's type." 3z + 7 and 3x + 7 are the same shape of object. The chapter switches between x, y and z within a single list precisely to make the letter look arbitrary.

Questions to check understanding

• State the degree of a given one-variable polynomial
• Name the family a polynomial belongs to, given the polynomial or given only its degree
• Write a polynomial to a specified degree, sometimes with a further condition on one coefficient — the End-of-Chapter Exercises on p. 36 open with exactly this, asking for degree 3 in x with the x² coefficient equal to –7
• Give the coefficient of a named power, including a power that is absent
• Give the constant term of a polynomial
• Decide whether a given expression qualifies as a one-variable polynomial at all

Examples worth working on the board

The chapter carries no figure in this stretch — it is all worked text and a short exercise set. Values marked verified are worked out here; the chapter prints no answers.

• The one-variable shortlist (p. 18). The chapter offers 4x, x² + 1, 2y – 5, 5y³ + y² + 2y – 1 and 3z + 7 as expressions in a single letter, and points out that the letter may be x, y or z — the choice of symbol is immaterial. Verified — of these five, three are degree 1 (4x, 2y – 5, 3z + 7), one is degree 2 (x² + 1) and one is degree 3 (5y³ + y² + 2y – 1). Useful as a sorting exercise.
• Reading the highest power (p. 18). In x² + 5x + 1 the chapter states the highest power of x is 2. In 5y³ + y² – 8 it states the highest power of y is 3. Note that the second has a gap where the y term would be.
• Coefficients of 5y³ + y² + 2y – 1 (p. 18). As printed: the coefficient of y³ is 5, of y² is 1, the coefficient of y is 2, and –1 sits as the constant term. Two traps live here. The coefficient of y² is 1 even though no digit is written, and the constant term is negative even though the expression shows a minus sign rather than a signed number.
• The four families (p. 18). 5y³ + y² + 2y – 1 has degree 3 and is called cubic. x² + 5x + 1 has degree 2 and is called quadratic. 3z + 7 has degree 1 and is called linear. The constant 8 has degree 0 and is called constant, and the chapter justifies this by writing it as 8x⁰, in which the power of x is 0.
• Why 8x⁰ is the whole argument. Verified — since x⁰ = 1 for any x other than 0, 8x⁰ and 8 take the same value everywhere the first is defined, so nothing about the number has changed; only its written form has. With the power made visible, the rule "read the largest power" returns 0 instead of failing, and the classification needs no special case. This is the reasoning to voice, and it is an added unpacking — the chapter states the rewrite and moves on.
• Exercise Set 2.1 (pp. 18–19), five items, all inputs to hand over:
  • Degrees of 2x² – 5x + 3, y³ + 2y – 1, – 9, and 4z – 3.
  • Write polynomials of degrees 1, 2 and 3 — an open-ended item, so any correct example works.
  • The coefficients of x² and x³ in x⁴ – 3x³ + 6x² – 2x + 7.
  • The coefficient of z in 4z³ + 5z² – 11.
  • The constant term of 9x³ + 5x² – 8x – 10.
• What the exercise items are testing. Verified — item 1(iii), the lone – 9, is the degree-0 case dressed as a throwaway. Item 3 asks for two coefficients in the opposite order to the way they appear, so the answers are 6 for x² and –3 for x³; reading them off in printed order is the error it is built to catch. Item 4 is the sharpest: 4z³ + 5z² – 11 has no z term written at all, so its coefficient is 0. Item 5 answers –10.
• Item 4 deserves a scene of its own. Verified — 4z³ + 5z² – 11 and 4z³ + 5z² + 0z – 11 are the same polynomial, and writing the second form makes the answer visible. "There is no z, so there is no coefficient" and "the coefficient is 0" are different statements, and only the second is right; the first would also make the degree-0 argument collapse.

Figures to have open

• No figure from the textbook is needed; this stretch of §2.1 prints none.
• A reusable annotation layer for a polynomial that can ring exponents, pull out coefficients one at a time, and insert a missing term with coefficient 0. Standard schematic, and the workhorse of the whole topic.
• A branching diagram from degree to family name, with one chapter example hanging under each branch. Standard schematic.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.1 "Introduction", p. 18 for the definitions and the four families, with Exercise Set 2.1 running from the foot of p. 18 to the top of p. 19.
• The two closing lines of §2.1, at the top of p. 19, restate that degree 1 means linear and announce the chapter's subject. They are the hinge into What makes a polynomial linear, and the equation you get by fixing its value.
• End-of-Chapter Exercises, item 1 on p. 36 and item 2 on p. 36, are the assessment forms of this topic; item 2(ii) evaluates 4t³ – t² + 6 at t = a, which quietly substitutes a letter for a letter.
• Chapter summary, pp. 39–40, restates the definitions of univariate polynomial, degree and linear polynomial with fresh examples.