The Language of Polynomials (VCE SSCE Mathematical Methods): Revision Notes

The Language of Polynomials

What is a polynomial function?

A polynomial function is a special type of mathematical function that can be written as a sum of terms, where each term consists of a coefficient multiplied by a power of $x$. The general form looks like this:

$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$

In this formula:

• $n$ is a non-negative integer (that is, $n \in \mathbb{N} \cup \{0\}$)
• The values $a_0, a_1, \ldots, a_n$ are real numbers called coefficients
• The coefficient $a_n$ of the highest power cannot be zero (unless the polynomial is the zero polynomial)

Important polynomial terminology

Zero polynomial: The number $0$ by itself is called the zero polynomial. It's a special case because it doesn't have a degree in the usual sense.

Leading term: The leading term is the term with the highest power of $x$ that has a non-zero coefficient. For example, in $P(x) = 3x^5 + 2x^2 - 7$, the leading term is $3x^5$.

Degree: The degree of a polynomial is the power (or index) of the leading term. Using our previous example, $P(x) = 3x^5 + 2x^2 - 7$ has degree $5$. We can write this as $\deg(P) = 5$.

Monic polynomial: A polynomial is called monic when its leading term has a coefficient of $1$. For instance, $P(x) = x^3 + 5x - 2$ is monic because the coefficient of $x^3$ is $1$, but $Q(x) = 2x^3 + 5x - 2$ is not monic.

Constant term: The constant term is the term that doesn't involve $x$ at all. It's the term with power $0$ (since $x^0 = 1$). In $P(x) = 4x^2 - 3x + 7$, the constant term is $7$.

Evaluating polynomials

To evaluate a polynomial at a particular value means to substitute that value for $x$ and calculate the result. This is written as $P(a)$, which means "the value of $P(x)$ when $x = a$".

The process is straightforward: replace every $x$ in the polynomial with the given value, then simplify using the order of operations.

Finding unknown coefficients

Sometimes we know the value of a polynomial at certain points, and we need to find unknown coefficients. This involves substituting the given value of $x$, forming an equation, and solving for the unknown.

When we have multiple unknowns, we need multiple equations. This often leads to solving simultaneous equations.

Arithmetic operations with polynomials

We can add, subtract, and multiply polynomials just like we do with numbers. The key is to work with like terms (terms with the same power of $x$).

Addition of polynomials

When adding polynomials, we combine like terms. Terms are "like" if they have the same power of $x$.

For example, if $P(x) = x^3 + 3x^2 + 2$ and $Q(x) = 2x^2 + 4$:

$$P(x) + Q(x) = (x^3 + 3x^2 + 2) + (2x^2 + 4)$$ $$= x^3 + 3x^2 + 2x^2 + 2 + 4$$ $$= x^3 + 5x^2 + 6$$

Notice that we simply added the coefficients of like terms: the $x^2$ terms combined to give $5x^2$, and the constant terms combined to give $6$.

Subtraction of polynomials

Subtraction works similarly, but remember to distribute the negative sign across all terms in the polynomial being subtracted.

Using the same polynomials:

$$P(x) - Q(x) = (x^3 + 3x^2 + 2) - (2x^2 + 4)$$ $$= x^3 + 3x^2 + 2 - 2x^2 - 4$$ $$= x^3 + x^2 - 2$$

Multiplication of polynomials

To multiply polynomials, we use the distributive property. Each term in the first polynomial must multiply each term in the second polynomial.

$$P(x) \times Q(x) = (x^3 + 3x^2 + 2)(2x^2 + 4)$$

We can distribute this step by step:

$$= (x^3 + 3x^2 + 2) \times 2x^2 + (x^3 + 3x^2 + 2) \times 4$$ $$= 2x^5 + 6x^4 + 4x^2 + 4x^3 + 12x^2 + 8$$

Now we collect like terms:

$$= 2x^5 + 6x^4 + 4x^3 + 16x^2 + 8$$

Rules for degrees

For polynomials $f$ and $g$ (both non-zero), we have useful rules about degrees:

Addition and subtraction:

$$\deg(f + g) \leq \max\{\deg(f), \deg(g)\}$$

The degree of the sum is at most the maximum of the two degrees. It could be less if the leading terms cancel out.

Multiplication:

$$\deg(f \times g) = \deg(f) + \deg(g)$$

The degree of the product equals the sum of the degrees. For example, multiplying a cubic (degree 3) by a quadratic (degree 2) always gives a polynomial of degree 5.

Equating coefficients

A powerful technique in polynomial work is equating coefficients. The principle is simple but extremely useful:

For example, if we know that:

$$4x^3 + 5x^2 - x + 3 = b_3 x^3 + b_2 x^2 + b_1 x + b_0$$

for all values of $x$, then we can immediately conclude:

• Coefficient of $x^3$: b₃ = 4
• Coefficient of $x^2$: b₂ = 5
• Coefficient of $x$: b₁ = -1
• Constant term: b₀ = 3

This technique is especially useful when we need to express a polynomial in a different form.

When equating coefficients shows impossibility

Sometimes, attempting to equate coefficients reveals that a polynomial cannot be written in a proposed form.