Polynomial Functions review games for Algebra 2.
This unit covers polynomial division, zeros and end behavior and fundamental theorem of algebra — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers polynomial division, zeros and end behavior and fundamental theorem of algebra — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Polynomial Division
Students must be able to divide polynomials using long division and synthetic division. Synthetic division only works when dividing by a linear factor of the form (x − c). The Remainder Theorem states that when a polynomial p(x) is divided by (x − c), the remainder equals p(c).
Key Points
- Use synthetic division when the divisor is (x − c); use long division for all other divisors
- Remainder Theorem: p(c) = remainder when p(x) is divided by (x − c)
- Factor Theorem: (x − c) is a factor of p(x) if and only if p(c) = 0
- The quotient degree is always one less than the dividend degree when dividing by a linear factor
Divide p(x) = 2x³ − 3x² + x − 5 by (x − 2) using synthetic division. Then state the remainder.
Set up synthetic division with c = 2 and coefficients 2, −3, 1, −5. Bring down 2, multiply 2×2=4, add to −3 to get 1, multiply 1×2=2, add to 1 to get 3, multiply 3×2=6, add to −5 to get 1. The quotient is 2x² + x + 3 with remainder 1, which means p(2) = 1 by the Remainder Theorem.
2 Zeros And End Behavior
The zeros of a polynomial are the x-values where the graph crosses or touches the x-axis, found by setting p(x) = 0. The multiplicity of a zero determines whether the graph crosses (odd multiplicity) or bounces off (even multiplicity) the x-axis. End behavior is determined solely by the leading term: its degree and leading coefficient.
Key Points
- Odd multiplicity → graph crosses the x-axis at that zero; even multiplicity → graph touches and turns around
- Even degree, positive leading coefficient: both ends go up (↑↑); negative leading coefficient: both ends go down (↓↓)
- Odd degree, positive leading coefficient: left end down, right end up (↓↑); negative: left end up, right end down (↑↓)
- A degree-n polynomial has at most n real zeros and at most n−1 turning points
For p(x) = −2x⁴(x − 3)²(x + 1), identify all zeros with multiplicities and describe end behavior.
The zeros are x = 0 (multiplicity 4, even → bounces), x = 3 (multiplicity 2, even → bounces), and x = −1 (multiplicity 1, odd → crosses). The leading term is −2x⁷ (degree 7, negative coefficient), so end behavior is: as x→−∞, y→+∞ and as x→+∞, y→−∞.
3 Fundamental Theorem Of Algebra
The Fundamental Theorem of Algebra states that every polynomial of degree n ≥ 1 has exactly n zeros in the complex number system, counting multiplicity. Complex zeros always come in conjugate pairs (a + bi and a − bi) when all coefficients are real. This means you can always fully factor a real-coefficient polynomial using linear and irreducible quadratic factors.
Key Points
- A degree-n polynomial has exactly n complex zeros (real + imaginary), counting multiplicity
- Complex zeros come in conjugate pairs: if (a + bi) is a zero, then (a − bi) is also a zero
- If you know one complex zero, you automatically know a second zero (its conjugate)
- Use known zeros to write factors, then divide to find remaining zeros
A degree-4 polynomial with real coefficients has zeros x = 2, x = −1, and x = 3i. Find all zeros and write the polynomial in factored form with leading coefficient 1.
Because 3i is a zero and the coefficients are real, its conjugate −3i must also be a zero — giving us all four zeros: 2, −1, 3i, −3i. Writing the factors: p(x) = (x − 2)(x + 1)(x − 3i)(x + 3i). The last two factors multiply to (x² + 9), so the fully factored form is p(x) = (x − 2)(x + 1)(x² + 9).
Questions, answered.
What is Polynomial Functions?
Polynomial Functions is Unit 4 of Algebra 2, covering polynomial division, zeros and end behavior and fundamental theorem of algebra.
How to study for Algebra 2 Unit 4?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.