Advanced Math • 35% of test
Polynomial Functions
Select your difficulty level to start practicing. We recommend mastering each level before moving to the next.
Beginner5 Sets
Beginner Practice
Start here to build your foundation. Clear problems and straightforward calculations.
Target Score
400-550
Intermediate5 Sets
Intermediate Practice
Level up with more complex equations and multi-step problems.
Target Score
550-700
Advanced3 Sets
Advanced Practice
Master the hardest concepts. Complex word problems and abstract applications.
Target Score
700-800
What is Polynomial Functions?
Polynomials include quadratics, cubics, and higher-degree expressions. The SAT tests factoring, finding zeros, and understanding end behavior based on degree and leading coefficient.
Step-by-Step Strategy
⚠️ Common Traps to Avoid
Frequently Asked Questions
What's the relationship between factors and zeros?
If (x - 3) is a factor, then x = 3 is a zero. Note the sign change!
How do I find zeros of a polynomial?
Factor completely and set each factor to zero. Or graph and find x-intercepts.
What determines end behavior?
Degree and leading coefficient. Odd degree: opposite ends. Even degree: same ends. Positive LC: up on right.
Can I use Desmos for polynomials?
Absolutely. Graphing to find zeros and relative extrema is a <a href='/math'>pro-level SAT Math strategy</a> that saves time.
What's the Factor Theorem?
If f(r) = 0, then (x - r) is a factor of f(x). This connects zeros and factors.
How do I multiply polynomials?
Use FOIL for binomials, or distribute each term systematically. Combine like terms at the end.
What is a 'Multiple Root' or 'Double Root'?
If a factor is squared (x - r)², the graph 'touches' the x-axis at r and turns back instead of crossing it.
How do I find the 'Remainder' without long division?
Use the Remainder Theorem: the remainder of f(x) divided by (x - a) is simply f(a). Plug the zero into the function.
What determines the number of 'Turning Points' in a graph?
A polynomial of degree n can have at most n-1 turning points (relative maxima or minima) on its graph.
How do I factor the 'Sum or Difference of Cubes'?
a³ + b³ = (a + b)(a² - ab + b²). a³ - b³ = (a - b)(a² + ab + b²). Use the SOAP mnemonic for signs.
What is 'Leading Coefficient Test' for end behavior?
If the leading term is axⁿ, then as x → ∞, f(x) follows the sign of a. Odd degree means opposite ends; even degree means same ends.
How do I interpret 'Multiplicity' of a zero in a graph?
Odd multiplicity (1, 3...) crosses the axis. Even multiplicity (2, 4...) 'bounces' off the axis at that specific zero.
Can a polynomial have zero real roots?
Yes, even-degree polynomials (like some quartics) can have zero real roots if they are shifted entirely above or below the x-axis.
What is 'Standard Form' for a polynomial?
Terms are ordered from highest degree (exponent) to lowest degree. This makes identifying 'degree' and 'LC' instant.
Is synthetic division allowed on the SAT?
Yes, it's a fast shortcut for dividing by factors. Practice these advanced division techniques in our <a href='/math/polynomial-functions/beginner'>Beginner Polynomial sets</a>.