Mathematics I · Week 4 — Polynomials

Week map

Polynomial notation

• $p(x) = a_n x^n + \cdots + a_1 x + a_0$ → polynomial in $x$ → coefficients real → $p(x) = 3x^3 - 2x + 7$.
• Degree $n$ → highest power with $a_n \neq 0$ → $5x^2 - 9$ has degree 2.
• Leading term $a_n x^n$ → dominates for large $|x|$ → end behaviour driver.
• Zero / root $r$ → $p(r) = 0$ → $(x-r)$ is a factor (over reals, if fully factorable).

Multiplicity

• $m=1$ → graph crosses axis at $r$.
• $m=2$ → often touches and turns (bounce).
• $m=3$ → crosses but flattens (flattened S).

Operations

• Add/subtract: combine like terms only ($x^2$ with $x^2$, not with $x$).
• Multiply: distribute each term; exponents add: $x^a \cdot x^b = x^{a+b}$.
• Long division: divide $p(x)$ by $(x-a)$; remainder degree less than divisor; if remainder 0, $(x-a)$ is factor.

Factor theorem

End behaviour

Pattern families

Easy — Degree and evaluation

• Combine polynomials; state degree and leading coefficient.
• Evaluate $p(2)$, $p(-1)$ by substitution.
• Identify $y$-intercept as $p(0) = a_0$.

Medium — Factor using theorem

• Test candidates; divide out linear factor; repeat.
• Build polynomial from given zeros with multiplicity in factored form.
• Expand factored form to standard when needed.

Hard — Graph reasoning

• Sketch sign chart from factorization.
• Relate multiplicity to touch/cross at each zero.
• Parameter $k$ so $x-a$ divides $p(x)-k$ (horizontal shift of roots).

Worked mini-examples

Traps

• Degree of zero polynomial handled specially; constant nonzero has degree 0.
• Dropping terms when adding — write all powers or use columns.
• Confusing zero of function with $y$-intercept ($x=0$ input).
• Assuming all even-multiplicity roots “bounce” without checking local shape — still need sign chart.
• Wrong end behaviour when only middle terms given — identify leading term first.

Diagnostic (try yourself)

1. What is the degree and leading coefficient of $p(x) = 4x - x^3 + 2x^5 - 7$?
2. If $p(x) = (x+2)^2(x-5)$, list all real zeros and their multiplicities.
3. Use the factor theorem to decide whether $(x+1)$ is a factor of $p(x) = x^3 + 3x^2 - x - 3$.
4. Multiply and simplify: $(x^2 - 3)(2x^2 + x - 4)$.
5. Describe end behaviour (left and right) of $f(x) = -3x^6 + 2x^2 - 1$ as $x \to \pm\infty$.