Chapter 2.2 — Exercise 2.1 — Types and Degree

Reading Off a Polynomial's Degree

Question 1 asks for the degree of six polynomials by finding the highest exponent-sum across all their terms. x⁵ − x⁴ + 3 has degree 5; 3x⁶ + 6y³ − 7 has degree 6, since 6 beats the y³ term's degree of 3; a lone constant like 5 always has degree 0.

The reliable method is to scan every term, compute each one's own exponent-sum, and keep only the largest value found — not the exponent of whichever term happens to be written first. 7 − x + 3x² illustrates why order can't be trusted: written left to right, the constant term comes first and the highest-degree term comes last, yet the degree is still 2 either way, because degree only cares about the largest exponent present, never about writing order. A single-variable term like −5xy² also deserves a careful look: its degree is 1 + 2 = 3, the sum across both variables in that one term, not just whichever variable's exponent looks biggest on its own.

One Variable, or More Than One?

Question 2 separates polynomials in one variable from those in two or more, plus a couple of non-polynomials mixed in to keep the check honest.

• One variable: 3x² − 2x + 5 and x² + 2 — a single symbol, x, runs through every term.
• More than one variable: p² − 3p + q and x¹⁰⁰ + y¹⁰⁰ — two distinct symbols appear, so "one variable" doesn't apply regardless of how clean the exponents look.
• Not a polynomial at all: y + 2/y (rewrites to y + 2y⁻¹, a negative exponent) and 5√x + x⁵ (the first term has a fractional exponent).

Notice that the third category is doing double duty — an expression can fail to be "a polynomial in one variable" for two entirely different reasons: either it isn't a polynomial at all (a negative or fractional exponent somewhere), or it is a polynomial but involves more than one variable. Sorting an expression into the right bucket means checking both conditions in order — first whether every exponent is a whole number, and only then, if that check passes, how many distinct variable symbols show up. Reversing that order causes real mistakes: judging "how many variables" before confirming the expression is even a polynomial can lead to calling something like y + 2/y "a one-variable expression," which is true in a narrow sense but skips past the more basic fact that it fails to be a polynomial at all.

Coefficients and the Degree-Based Names

Question 3 asks for the coefficient of x³ across eight polynomials — the number multiplying that specific power, or 0 if x³ doesn't appear in the expression at all. In 2x² + 5, for instance, there's no x³ term whatsoever, so its coefficient is taken as exactly 0, not left undefined. This matters more than it first appears, because treating a missing term as "coefficient 0" rather than "no information" is exactly what makes the standard-form template from the introduction — a₀xⁿ + a₁xⁿ⁻¹ + … + aₙ — apply cleanly to every polynomial without exception, gaps included.

Question 4 then classifies six polynomials by degree:

• Linear (degree 1): x − 1, 3p.
• Quadratic (degree 2): 5x² + x − 7, x² + x + 4, πr².
• Cubic (degree 3): x − x³.

πr² is worth pausing on: despite the irrational constant π sitting in front, the exponent on r is a plain 2, so the polynomial is quadratic — degree only ever looks at the variable's exponent, never at whether the coefficient itself happens to be a whole number. It's a mistake worth flagging explicitly, because it's tempting to see an unfamiliar symbol like π sitting inside a polynomial and assume something unusual is going on — nothing is; π is simply playing the same role here that 5 or −7 would play anywhere else.

True or False, With a Reason Each Time

Question 5 tests conceptual precision rather than calculation:

• A binomial has two terms — true, by definition, regardless of what degree those two terms happen to be.
• Every polynomial is a binomial — false; a polynomial can have any number of terms, one or more.
• A binomial may have degree 3 — true; term count and degree are unrelated, so a two-term expression like x³ + 1 is a perfectly valid cubic binomial.
• The degree of the zero polynomial is zero — false; it's undefined, not zero, since 0 can be written as 0×xⁿ for literally any n.
• The degree of x² + 2xy + y² is 2 — true; the highest exponent-sum across all three terms is 2.
• πr² is a monomial — true; it has exactly one term, and term count is all that "monomial" is measuring.

Six statements, six separate justifications — none of them solved by calculation, all of them solved by returning to the exact wording of a definition and checking it carefully against the example at hand.

Building Examples in Reverse

The final question flips the process around: instead of classifying a given polynomial, it asks for one to be constructed to a specification. A monomial of degree 10 works out to something like 9x²y⁸ (exponents 2 and 8 sum to 10, and there's exactly one term). A trinomial of the same degree, such as 5 − x³ − 8x¹⁰, needs to keep exactly three non-zero terms while still reaching degree 10 through its highest term. Working from a definition to a matching example, rather than the other way around, is a genuinely different skill from classification — it's easy to correctly classify a polynomial someone else wrote, and considerably harder to construct one from scratch that satisfies several conditions simultaneously. A useful way to build such an example methodically, rather than by trial and error, is to fix the term count first, then choose exponents for each term that sum correctly for at least one of them to hit the required degree — the other terms just need to stay at or below that degree, and nothing stops two different terms from sharing the same degree as each other.

What Carries Forward From Here

The classification fluency built in this exercise gets used constantly for the rest of the chapter — every zero-finding and factorisation problem from here on starts by silently registering a polynomial's degree and type before doing anything else with it — knowing in advance that a polynomial is quadratic, for instance, is what tells you to expect at most two zeroes rather than search indefinitely for a third. Exercise 2.2 is the next step, asking what value of x actually makes one of these polynomials equal zero.