Define a polynomial and use the language of degree, leading term, leading coefficient, monic and the zero polynomial, including the degree of sums and products and equating coefficients of identically equal polynomials

What this dot point is asking

This is your first formal contact with polynomials in Extension 1. NESA wants you to be fluent in the vocabulary before you start dividing, factoring and graphing them: what a polynomial actually is, its degree, its leading term and leading coefficient, what monic means, and the special status of the zero polynomial. On top of the vocabulary, two working skills are examined directly: predicting the degree of a sum or a product without expanding fully, and finding unknown coefficients by equating two polynomials that are identically equal.

The answer

What a polynomial is

A polynomial in $x$ is a sum of terms, each a number times a whole-number power of $x$:

where the coefficients $a_n, a_{n-1}, \dots, a_0$ are real numbers and the powers are non-negative integers. The two conditions that do the work are hidden in that sentence. The powers must be whole numbers, so $\sqrt{x} = x^{1/2}$, $\dfrac{1}{x} = x^{-1}$ and $2^x$ are all banned: none is a polynomial. And the coefficients are just constants, so $3x^2 - 7x + 1$ qualifies but $x^2 + x \log_e x$ does not.

Degree, leading term and leading coefficient

The vocabulary all keys off the term of highest power that actually appears.

• The leading term is the term of highest power with a non-zero coefficient.
• The degree is the power of the leading term.
• The leading coefficient is the coefficient of the leading term.
• The constant term is $a_0 = P(0)$, the term with no $x$.
• A monic polynomial is one whose leading coefficient is $1$.

So for $P(x) = 3x^4 - 5x^2 + 2x - 7$ the leading term is $3x^4$, the degree is $4$, the leading coefficient is $3$ (so it is not monic), and the constant term is $-7$.

Low degrees have names you should use: degree $0$ is a (non-zero) constant, degree $1$ is linear, degree $2$ is quadratic, degree $3$ is cubic, degree $4$ is quartic, degree $5$ is quintic.

There is one trap NESA likes here. A constant function is a linear function, but a constant polynomial is not a linear polynomial: a linear polynomial has degree $1$, while a non-zero constant has degree $0$. The word linear means degree exactly one when it is applied to polynomials.

The zero polynomial

The constant polynomial $a_0$ has degree $0$ as long as $a_0 \ne 0$. The single exception is the zero polynomial $Z(x) = 0$. It has every coefficient equal to zero, so it has no leading term, and therefore it has no degree at all. It is not "degree $0$"; it is the one polynomial with no degree. This sounds like a technicality, but it is exactly the case that breaks the degree rules below, which is why the rules are stated with care.

The degree of a sum or difference

Add or subtract two polynomials and you get another polynomial, formed by adding or subtracting matching coefficients. The degree usually behaves simply, but there is a catch when the two have the same degree.

Let $P(x)$ and $Q(x)$ be non-zero polynomials of degrees $n$ and $m$.

• If $n \ne m$, then $\deg\big(P(x) \pm Q(x)\big)$ is the larger of $n$ and $m$. The bigger leading term has nothing to cancel against.
• If $n = m$, the leading terms can cancel, so $\deg\big(P(x) \pm Q(x)\big) \le n$, and the result might even be the zero polynomial (with no degree).

For example, $(x^2 - 3x + 2) + (-x^2 + 4x + 9) = x + 11$: two quadratics whose leading terms cancel, leaving a polynomial of degree $1$. And $P(x) + \big(-P(x)\big) = 0$ for every $P$, where the opposite polynomial $-P(x)$ is formed by negating every coefficient; the sum is the zero polynomial, which is why "no degree" has to be allowed.

The degree of a product

Multiplication is the well-behaved case. The leading term of a product is the product of the two leading terms, and multiplying two non-zero numbers can never give zero, so the top term never disappears. Hence the degree of a product is the sum of the degrees:

This lets you read off the degree, the leading coefficient and the constant term of a big product without expanding it. For $(2x^2 + 1)(x^3 - 4)(5x^5 + 3)$ the degree is $2 + 3 + 5 = 10$, the leading coefficient is $2 \times 1 \times 5 = 10$, and the constant term is $1 \times (-4) \times 3 = -12$.

Identically equal polynomials and equating coefficients

Two polynomials $P(x)$ and $Q(x)$ are identically equal, written $P(x) \equiv Q(x)$, if they are equal for every value of $x$, not just for a few. The single most useful fact in this whole topic follows from that:

If two polynomials are identically equal, then their corresponding coefficients are equal.

This is what powers "equating coefficients": you write the same polynomial two different ways, set the coefficient of each power on one side equal to the coefficient of that power on the other, and solve the resulting equations for the unknowns. It is the standard method for splitting a fraction into partial fractions later in the course, and it turns one polynomial identity into a whole system of ordinary equations.

The distinction between $\equiv$ (true for all $x$) and $=$ (true for the particular $x$ you are solving for) matters. An equation like $x^2 - 5x + 6 = 0$ is true only for $x = 2$ and $x = 3$; an identity like $(x - 2)(x - 3) \equiv x^2 - 5x + 6$ is true for every $x$, and only an identity lets you equate coefficients.

Even and odd polynomials

A function is even if $P(-x) = P(x)$ for all $x$ (symmetric in the $y$-axis) and odd if $P(-x) = -P(x)$ for all $x$ (symmetric in the origin). For a polynomial these reduce to a clean statement about which powers appear, because replacing $x$ with $-x$ flips the sign of every odd power and leaves every even power alone:

• a polynomial is even exactly when only even powers have non-zero coefficients (for example $3x^4 - 5x^2 + 2$);
• a polynomial is odd exactly when only odd powers have non-zero coefficients (for example $2x^5 - 3x^3 + x$).

So if $P(x) = a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$ is even, then comparing $P(-x)$ with $P(x)$ forces $a_3 = a_1 = 0$; if it is odd, the even-power coefficients $a_4, a_2, a_0$ are all zero. Most polynomials are neither even nor odd.

How exam questions ask about the language of polynomials

The wording is predictable once you know the moves to match it to.

• "State the degree, leading coefficient, leading term and constant term." Read them straight off the highest-power term, after first expanding if the polynomial is given in factored or bracketed form. Watch a leading coefficient that is negative or fractional, and remember monic means leading coefficient exactly $1$.
• "What is the degree of $P(x) + Q(x)$ / $P(x)Q(x)$?" For a product, add the degrees, no expansion needed. For a sum, take the larger degree, unless the two degrees are equal, in which case warn that the leading terms might cancel and the degree could drop.
• "Find the constants so that ... $\equiv$ ... for all $x$." Equate coefficients of each power to get a system of equations, then solve. The phrase "for all $x$" is the signal that this is an identity, not an equation to be solved for $x$.
• "Show that $b = d = 0$ if $P(x)$ is even" (or the odd version). Write out $P(-x)$, set it identically equal to $P(x)$ (or to $-P(x)$), and equate coefficients; the odd-power (or even-power) coefficients are forced to zero.
• "Give an example of two polynomials of degree $2$ whose sum has degree $0$." This is testing whether you understand same-degree cancellation: choose two quadratics whose $x^2$ and $x$ terms cancel but whose constants do not, such as $x^2 + 1$ and $-x^2 + 1$, summing to $2$.