Introduction to Polynomials

A polynomial in $x$ is an expression of the form:

where $n$ is a non-negative integer (the degree), each $a_i$ is a constant coefficient, and $a_n \ne 0$ (so the degree is exactly $n$).

Examples of polynomials:

• $3x^2 - 2x + 5$ — degree 2, leading coefficient 3
• $x^4 - 1$ — degree 4, leading coefficient 1 (monic)
• $7$ — degree 0, a constant polynomial

Not polynomials:

• $x^2 + \dfrac{1}{x} = x^2 + x^{-1}$ — negative power $(-1)$
• $\sqrt{x} + 2 = x^{1/2} + 2$ — fractional power $(\frac{1}{2})$
• $2^x$ — variable in the exponent, not the base

Polynomial: P(x) = a_n x^n + + a_0 where each power is a non-negative integer; Degree = highest power with a non-zero coefficient

Pause — copy the polynomial definition into your book: $P(x) = a_n x^n + \cdots + a_1 x + a_0$ where each power is a non-negative integer; degree = highest power with a non-zero coefficient.

We just saw that a polynomial $P(x) = a_n x^n + \cdots + a_0$ has degree $n$ (highest non-zero power). That raises a question: to check whether a given value $x = a$ satisfies $P(x) = 0$, you need to evaluate $P(a)$ — what is the safest method to avoid sign errors? This card answers it → substitute using brackets around the value, evaluate each term separately, then combine.

To evaluate a polynomial at a specific value, substitute the value for $x$ and simplify step by step. Do not skip steps — sign errors are the main source of lost marks.

Example: If $P(x) = 2x^3 - x^2 + 3x - 4$, find $P(2)$.

You can also use evaluation to find unknown coefficients. If $P(x) = x^3 - 2x^2 + kx + 5$ and $P(2) = 11$:

Substitute x = a and evaluate each term separately before combining; Use brackets around substituted values: write 2(2)^3 not 22^3

Pause — copy the evaluation rule into your book: substitute using brackets — write $2(2)^3$ not $22^3$; evaluate each term separately before combining to avoid sign errors.