Section 3.poly

Polynomials

Polynomials are a major topic on the final — expect 3–4 questions on end behavior, zeros and multiplicity, and synthetic division. Being able to factor a polynomial and describe its graph is one of the core skills the exam tests.

⚡ Quick Summary

Polynomials are expressions built from adding, subtracting, and multiplying $x$ by itself — things like $x^3 - 4x + 1$ . Their graphs are smooth, continuous curves with no sharp corners or breaks. The degree (the highest power of $x$ ) controls the overall shape: higher degrees mean more possible wiggles and turns.

End Behavior (Leading Term Test)

f(x) = a_n x^n + \cdots \implies \text{end behavior matches } a_n x^n

Even degree → both ends same direction; odd degree → opposite ends

Division Algorithm

f(x) = d(x) \cdot q(x) + r(x)

dividend = divisor × quotient + remainder (degree of r < degree of d)

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The end behavior — what happens at the far left and far right of the graph — is controlled entirely by the leading term (the highest-power piece). Think of it like zooming out on a map: the small bumps disappear and only the big trend remains. Even-degree polynomials have both ends going the same direction (both up or both down), while odd-degree polynomials go in opposite directions.

Zeros (also called roots) are where the graph crosses or touches the $x$ -axis. Each zero has a multiplicity — how many times that factor repeats. If the multiplicity is odd, the graph crosses straight through the axis at that point. If it's even, the graph just touches and bounces back, like a ball bouncing off the ground.

To find zeros, you often start by listing candidates using the Rational Zero Theorem, then test them one by one using synthetic division (a quick shorthand for polynomial division). Once you find one zero, you work with a simpler polynomial and keep going until it's fully factored.

Key Formulas

End Behavior (Leading Term Test)

f(x) = a_n x^n + \cdots \implies \text{end behavior matches } a_n x^n

Even degree → both ends same direction; odd degree → opposite ends

Division Algorithm

f(x) = d(x) \cdot q(x) + r(x)

dividend = divisor × quotient + remainder (degree of r < degree of d)

Remainder Theorem

f(c) = \text{remainder when } f(x) \div (x - c)

Evaluate f(c) without plugging in — just read the remainder from synthetic division

Factor Theorem

(x - c) \text{ is a factor of } f(x) \iff f(c) = 0

Rational Zero Theorem

\text{Possible rational zeros} = \pm\frac{p}{q}

p = factors of the constant term, q = factors of the leading coefficient

Key Takeaways

✓End behavior depends only on the leading term: even degree → both ends same direction, odd degree → opposite ends
✓Zeros with odd multiplicity cross the $x$ -axis; zeros with even multiplicity touch and bounce
✓Remainder Theorem: $f(c)$ equals the remainder when dividing $f(x)$ by $(x - c)$ ; if the remainder is $0$ , then $(x - c)$ is a factor
✓Rational Zero Theorem: possible rational zeros are $\pm\frac{\text{factors of constant}}{\text{factors of leading coeff}}$

⚠️ Common Mistakes

✗Forgetting to use $0$ as a placeholder for missing terms in synthetic division — if there's no $x^3$ term, you still need a $0$ in that slot or everything after shifts
✗Confusing end behavior with local behavior — end behavior depends ONLY on the leading term $a_n x^n$ , no matter what the middle of the graph looks like
✗Saying a zero with even multiplicity "crosses" the axis — even multiplicity means the graph TOUCHES and turns around; odd multiplicity is what crosses
✗After a successful synthetic division, testing the next candidate against the original polynomial instead of the reduced quotient — always continue with the quotient

Next: Section 3.7 Builds on: 3.cq

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3.cq Complex Numbers & Quadratics