Quick Answer
Zeros of a function are the input values where f(x) equals zero. On the Digital SAT, this concept is a staple of the Advanced Math section, typically appearing in 3–5 questions per exam. Students must identify these as x-intercepts on a graph or solve for them algebraically using factoring or the quadratic formula.
The zeros of a function, also known as roots or x-intercepts, are the values of x such that f(x) = 0. Algebraically, if a polynomial has a zero at x = k, then (x - k) is a factor of that polynomial.
Question: A function is defined as f(x) = (x + 3)(x - 5). What is the sum of the zeros of function f? Solution: The zeros occur when f(x) = 0. Setting (x + 3)(x - 5) = 0 gives x = -3 and x = 5. The sum of these zeros is -3 + 5 = 2.
Sign Errors: Students often mistake the factor (x - k) for a zero at -k instead of +k because they forget that the zero is the value that makes the factor equal zero.
Confusing Zeros with Y-intercepts: Some test-takers calculate f(0) instead of solving for f(x) = 0 because they confuse the 'zero' of the input with the 'zero' of the output.
Overlooking Multiplicity: Students might count a tangent point where the graph touches the x-axis as one zero but fail to recognize it represents a squared factor in the equation.
Students targeting 750+ should know that the sum of the zeros of a quadratic ax² + bx + c can be found instantly using -b/a, and the product using c/a, which saves critical time on the Digital SAT when the equation is not easily factorable.
Quadratic Equation
A quadratic equation is a second-degree polynomial equation typically written in standard form as ax² + bx + c = 0. On the Digital SAT, these equations appear frequently in the Advanced Math section, accounting for approximately 15% of math questions. Students must solve them using factoring, completing the square, or the quadratic formula.
Polynomial
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.
Factoring
Factoring is the mathematical process of breaking down a polynomial into a product of simpler expressions or factors. On the Digital SAT, this technique is frequently tested in the Math modules, appearing in approximately 10-15% of algebra and advanced math questions, often requiring students to identify equivalent expressions or find the zeros of quadratic functions.
Roots
Roots are the input values that make a function equal zero. On the Digital SAT, roots appear frequently in the Math section, especially within quadratic and polynomial problems. They are typically tested as x-intercepts on a graph or as solutions to equations, appearing in approximately 15% of Advanced Math questions.
X-Intercept
An x-intercept is the point where a graph crosses the horizontal axis on the Digital SAT. This concept appears frequently in Math Modules 1 and 2, often within linear or quadratic modeling questions. At this point, the y-value is always zero, representing a critical solution or root of the function.
Zeros of a function on the SAT refer to the x-values where the output of a mathematical expression equals zero. These are also known as roots or x-intercepts. On the Digital SAT, you will encounter these in the Math section, specifically within algebra and advanced math modules, where you must identify them from equations, tables, or graphs to solve for unknown constants or analyze polynomial behavior.
To calculate the zeros of a function f(x), set the entire expression equal to zero and solve for x. For quadratic functions, you can often find zeros by factoring the expression into (x-p)(x-q) or by using the quadratic formula. On the Digital SAT, using the built-in Desmos graphing calculator to find x-intercepts is an extremely efficient strategy for complex polynomials.
There is no functional difference between zeros of a function and x-intercepts; they describe the same mathematical concept from different perspectives. 'Zeros' or 'roots' usually refer to the algebraic solutions to the equation f(x) = 0, while 'x-intercepts' refer to the specific points (x, 0) where the graph of the function crosses the horizontal axis. On the SAT, these terms are used interchangeably to test your understanding of polynomial graphs.
While the exact number varies by test form, you can typically expect to see approximately 3 to 5 questions that directly or indirectly test zeros of a function. These questions range from simple identification of intercepts on a provided graph to more complex problems involving the discriminant or the relationship between factors and coefficients in higher-degree polynomials in the Math modules.
