Algebra II · Polynomial Functions

Zeros, Roots &
End Behavior

Master polynomial functions — writing equations with conjugate pairs, describing end behavior in proper notation, and reading graph behavior.

20Questions
4Topics
0Score
⚡ Quick Reference

Key Rules at a Glance

End Behavior — Even Degree
Both ends go the same direction. Positive leading → both ↑. Negative leading → both ↓.
End Behavior — Odd Degree
Ends go opposite directions. Positive: left ↓, right ↑. Negative: left ↑, right ↓.
Complex Conjugate Pairs
If \(a+bi\) is a zero → \(a-bi\) must also be a zero. Always come in pairs when coefficients are rational.
Irrational Conjugate Pairs
If \(a+\sqrt{b}\) is a zero → \(a-\sqrt{b}\) must also be a zero (rational coefficients required).
Writing from Zeros
\(f(x)=(x-r_1)(x-r_2)\cdots(x-r_n)\), then expand fully. Degree = total number of zeros.
Multiplicity Rule
Even multiplicity → graph touches x-axis & bounces. Odd multiplicity → graph crosses through.
🔑 SODS · Same=Even · Opposite=Odd · Direction=Leading · Conjugate Pairs Always Travel Together
Topic 01 · End Behavior
Q 01
End Behavior · Odd Degree · Positive Leading
Easy
What is the end behavior of \(f(x)=x^3+10x^2+32x+34\)?
🧠 KEYWORD: "ODD = OPPOSITE ENDS" Odd degree (3) → left and right go in opposite directions. Positive leading (+1) → right rises to \(+\infty\), left falls to \(-\infty\).
Q 02
End Behavior · Even Degree · Negative Leading
Easy
Describe the end behavior of \(f(x)=-x^2-8x-15\).
🧠 KEYWORD: "EVEN = SAME ENDS" Degree 2 (even) → both ends go the same direction. Leading coefficient is \(-1\) (negative) → both ends fall to \(-\infty\).
Q 03
End Behavior · Even Degree · Negative Leading
Easy
What is the end behavior of \(f(x)=-x^4+x^2+2\)?
🧠 TRICK: Only the HIGHEST degree term controls end behavior! Ignore \(+x^2+2\). Focus on \(-x^4\): even degree + negative leading → both ends fall to \(-\infty\).
Q 04
End Behavior · Even Degree · Positive Leading
Easy
Describe the end behavior of \(f(x)=x^4-4x^2-x+3\).
Q 05
End Behavior · Odd Degree · Negative Leading
Medium
What is the end behavior of \(f(x)=-3x^5+2x^3-x\)?
🧠 KEYWORD: "Negative Odd = Falls Right, Rises Left" Odd degree (5) → opposite ends. Negative leading (\(-3\)) → right end falls to \(-\infty\), left end rises to \(+\infty\).
Topic 02 · Writing Polynomials from Zeros
Q 06
Writing Polynomials · Complex Conjugate Pair
Medium
Write \(f(x)\) of least degree, leading coefficient 1, rational coefficients, zeros: \(2,\;-3,\;-3i\).
🧠 CONJUGATE RULE: \(-3i\) given → \(+3i\) is also required! Four zeros total: \(2,\;-3,\;-3i,\;+3i\) → degree 4.
Step-by-step Step 1: \((x+3i)(x-3i)=x^2-(3i)^2=x^2-(-9)=x^2+9\)
Step 2: \((x-2)(x+3)=x^2+x-6\)
Step 3: \((x^2+x-6)(x^2+9)=x^4+x^3+3x^2+9x-54\)
Q 07
Writing Polynomials · Three Real Zeros
Easy
Write \(f(x)\) of least degree, leading coefficient 1, with zeros \(-1,\;2,\;4\).
🧠 RULE: Zero \(r\) → factor \((x-r)\). Watch the sign! Zero \(-1\) → \((x+1)\). Zero \(2\) → \((x-2)\). Zero \(4\) → \((x-4)\).
Step-by-step \((x+1)(x-2)=x^2-x-2\)
\((x^2-x-2)(x-4)=x^3-5x^2+2x+8\)
Q 08
Conjugate Pairs · Irrational Zeros
Easy
A polynomial with rational coefficients has \(\sqrt{5}\) as a zero. Which other value must also be a zero?
🧠 IRRATIONAL CONJUGATE: \(\sqrt{b}\) always pairs with \(-\sqrt{b}\)! Without the conjugate partner, the polynomial would have irrational coefficients. \(\sqrt{5}\) and \(-\sqrt{5}\) together give factor \((x^2-5)\) with rational coefficients.
Q 09
Writing Polynomials · Repeated Zero (Multiplicity 2)
Medium
Write \(f(x)\) of least degree, leading coefficient 1, with zero \(0\) (multiplicity 2) and zero \(3\).
🧠 MULTIPLICITY: multiplicity 2 → write the factor squared \((x-r)^2\)! Zero 0, mult. 2 → \(x^2\). Zero 3 → \((x-3)\). Expand: \(x^2(x-3)=x^3-3x^2\).
Q 10
Writing Polynomials · Two Complex Conjugate Pairs
Hard
Write \(f(x)\) of least degree, leading coefficient 1, rational coefficients, zeros \(4i\) and \(-2i\).
🧠 Each complex zero brings its conjugate — two pairs = degree 4! \(4i\) → also \(-4i\).  \(-2i\) → also \(+2i\). Four zeros total.
\((x^2+16)(x^2+4)=x^4+20x^2+64\)
Topic 03 · Finding Zeros from Equations
Q 11
Finding Zeros · Factoring a Quadratic
Easy
Find all zeros of \(f(x)=x^2+8x+15\).
🧠 Find two numbers that MULTIPLY to +15 and ADD to +8. \(+3\) and \(+5\): multiply to 15, add to 8. So \((x+3)(x+5)=0\) → \(x=-3,\;x=-5\).
Q 12
Finding Zeros · Sum of Squares → Imaginary Roots
Medium
Find all zeros of \(f(x)=x^2+9\).
🧠 Sum of squares NEVER factors over the reals — always imaginary! \(x^2+9=0\;\Rightarrow\;x^2=-9\;\Rightarrow\;x=\pm\sqrt{-9}=\pm 3i\)
Q 13
Fundamental Theorem · Degree = Number of Zeros
Easy
How many zeros (counting multiplicity) does \(f(x)=x^5-3x^3+x\) have in the complex number system?
🧠 FUNDAMENTAL THEOREM: Degree \(n\) = exactly \(n\) complex zeros! The highest power is 5 → exactly 5 zeros in the complex number system (some may be imaginary or repeated).
Q 14
Finding Zeros · Difference of Squares Applied Twice
Medium
Find all zeros of \(f(x)=x^4-16\).
🧠 DOUBLE DIFFERENCE OF SQUARES — factor twice! \(x^4-16=(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)\)
Then \(x^2+4=0\;\Rightarrow\;x=\pm 2i\). All four zeros: \(2,\;-2,\;2i,\;-2i\).
Q 15
Multiplicity · Graph Behavior at a Zero
Hard
For \(f(x)=(x-1)^2(x+3)\), what does the graph do at \(x=1\)?
🧠 MULTIPLICITY RULE: Even → Touch & Bounce. Odd → Cross Through. Zero \(x=1\) has multiplicity 2 (even) → graph touches x-axis and bounces back. Does NOT cross.
Topic 04 · Mixed Challenge
Q 16
Identifying Leading Term from End Behavior
Medium
Which polynomial has: as \(x\to+\infty,\;f\to+\infty\) AND as \(x\to-\infty,\;f\to-\infty\)?
🧠 "Rises right, falls left" = POSITIVE LEADING + ODD DEGREE. Opposite ends → odd. Right rises → positive leading. The only match is \(x^3\) (degree 3, positive).
Q 17
Complex Numbers · Powers of \(i\)
Hard
Simplify: \(i^{23}\)
🧠 Powers of \(i\) cycle every 4: \(i^1=i,\;i^2=-1,\;i^3=-i,\;i^4=1\), repeat! Divide by 4 and check the remainder: \(23\div 4=5\) remainder \(\mathbf{3}\) → \(i^{23}=i^3=-i\)
Q 18
Minimum Degree from Given Zeros
Medium
A polynomial has rational coefficients and zeros \(3,\;i,\;\sqrt{2}\). What is the minimum possible degree?
🧠 Count ALL zeros including forced conjugate partners! \(3\) (rational real) → 1 zero, no partner.  \(i\) → forces \(-i\) → 2 zeros.  \(\sqrt{2}\) → forces \(-\sqrt{2}\) → 2 zeros.
Total: \(1+2+2=5\) → minimum degree = 5.
Q 19
Graph Behavior → Leading Term
Hard
A polynomial graph falls to the left AND falls to the right. Which could be its leading term?
🧠 "Falls BOTH sides" = EVEN degree + NEGATIVE leading! Same direction (both down) → even degree. Both ends go to \(-\infty\) → negative leading coefficient.
Q 20
Full Review · Zeros + End Behavior Combined
Hard
\(f(x)\) has rational coefficients, leading coefficient 1, and zeros \(-1\) and \(2i\). What is the end behavior?
🧠 Step 1: Find ALL zeros first, THEN determine degree! \(-1\) (rational real) = 1 zero.  \(2i\) → forces \(-2i\) = 2 more zeros.
Total: 3 zeros → degree 3 (ODD) + positive leading → rises right, falls left.
0/20 Correct