Week 4 — algebra of polynomials

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Week map

Classify → Represent → Execute → Trap-check

• Recognize: Ask: What does factor theorem say in one line?
• Procedure: Combine like terms to standard form. State degree and leading coefficient. Evaluate $p(a)$ by substitution—this is the hook for factor checks.
• Variations / traps: Watch for: Dropping terms when adding polynomials.

Formula chain (compressed)

polynomial ops → f(a)=0 ⇔ (x−a) factor → multiplicity → end behaviour from degree.

1. Factor theorem — f(a)=0 ⇔ (x−a) | f(x) — root ↔ factor
2. Remainder — remainder ÷ (x−a) is f(a) — synthetic division check
3. Degree — highest power of x with nonzero coeff — classify polynomial
4. Multiplicity — (x−a)^k factor → k — touch vs cross at root
5. End behaviour — even degree: same ends; odd: opposite — leading coeff sign sets direction

Deep study

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 $x \to \pm\infty$.

ChatGPT prep archive

• Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

• Polynomial: sum of terms $a_n x^n$; degree is highest power with nonzero coefficient.
• Operations: add/subtract like terms; multiply by distributing; long division for $p(x)/q(x)$.
• Factor theorem: $(x-a)$ is a factor iff $p(a)=0$; multiplicity counts repeated roots.
• End behaviour: dominated by leading term $a_n x^n$; sign of $a_n$ and parity of $n$ set both ends.

Notation & vocabulary

Pattern families

Easy — Evaluate and classify degree

Medium — Factor using theorem

Hard — Sketch from factorization

Drill these on the pattern atlas — filter to week 4.

Traps

• Dropping terms when adding polynomials.
• Confusing zero of function with $y$-intercept (evaluate $p(0)$).
• Single crossing at a double root—multiplicity 2 often touches axis.
• Wrong end behaviour when degree is even and leading coefficient negative.

Retrieval prompts

1. What does factor theorem say in one line?
2. How does multiplicity 3 affect graph at a zero?
3. What determines end behaviour of a polynomial?

Practice loop

1. Read Deep study (if present) or core concepts once.
2. Recite the formula chain without looking.
3. Open one easy pattern on the interactive atlas for week 4.
4. Attempt without solutions; mark studied after an honest try.
5. Say one trap aloud before closing the tab.

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk