Quick Answer
A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. On the Digital SAT, polynomials frequently appear in the Advanced Math section, typically requiring students to add, subtract, multiply, or factor expressions. These questions often represent approximately 10-15% of the math content across both modules.
A polynomial is an algebraic expression that is a sum of terms in the form $ax^n$, where $a$ is a real number coefficient and $n$ is a non-negative integer exponent. For example, $f(x) = 3x^2 - 5x + 7$ is a standard quadratic polynomial.
Question: Which of the following is equivalent to $(2x^2 + 3x - 4) - (x^2 - 5x + 2)$? Solution: Distribute the negative sign: $2x^2 + 3x - 4 - x^2 + 5x - 2$. Group like terms: $(2x^2 - x^2) + (3x + 5x) + (-4 - 2) = x^2 + 8x - 6$.
Failing to distribute the negative sign: When subtracting polynomials, students often forget to flip the signs of every term within the second set of parentheses.
Incorrect exponent rules: Students sometimes add exponents when adding terms (e.g., x^2 + x^2 = x^4) instead of combining coefficients (x^2 + x^2 = 2x^2).
Confusing factors with roots: Students may incorrectly identify the root of (x + 3) as x = 3 instead of x = -3.
Students targeting 750+ should know that the Remainder Theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a), which allows for rapid evaluation of constants in complex polynomial equations without full long division.
Leading Coefficient
The leading coefficient is the numerical factor of the term with the highest degree in a polynomial. On the Digital SAT, this concept frequently appears in the Advanced Math section, particularly within questions regarding parabola orientation and polynomial end behavior. It is essential for identifying the direction of a graph's opening.
End Behavior
End behavior describes the direction of a function's graph as the input value approaches positive or negative infinity. On the Digital SAT, this concept is tested within the Advanced Math category, appearing approximately 1–3 times per exam. It requires students to predict whether a graph rises or falls based on its leading coefficient and degree.
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in an algebraic expression. On the Digital SAT, this concept is frequently tested within the 'Passport to Advanced Math' section. It typically determines the maximum number of x-intercepts and overall end behavior, appearing in approximately 2-4 questions per test.
A polynomial on the SAT is an expression involving variables raised to whole-number powers, such as $x^2 + 2x + 1$. It appears primarily in the Advanced Math section of the Digital SAT. Students are expected to manipulate these expressions through addition, subtraction, and multiplication, or use them to model real-world scenarios. Mastery of polynomials is essential for solving approximately 15% of the math section.
To identify a polynomial, ensure that all variables have exponents that are non-negative integers (0, 1, 2, ...). An expression is not a polynomial if it contains a variable in the denominator, a variable under a radical (like $\sqrt{x}$), or a negative exponent (like $x^{-1}$). On the SAT, you will mostly work with linear, quadratic, and cubic polynomials in standard or factored form.
A polynomial is a type of mathematical expression, whereas an equation is a statement that two expressions are equal. For example, $3x^2 + 2x$ is a polynomial expression, but $3x^2 + 2x = 0$ is a polynomial equation. On the SAT, you may be asked to simplify a polynomial expression or solve a polynomial equation for its roots or x-intercepts.
While the exact number varies by test version, the Digital SAT typically includes approximately 4 to 6 questions directly involving polynomial manipulation or functions. These are categorized under 'Advanced Math.' Because polynomials form the basis for quadratics and other higher-order functions, their underlying principles are relevant to a significant portion of the overall Math score across both modules.
