Quick Answer
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.
The leading coefficient is the coefficient of the variable with the highest exponent in a polynomial expression. In the standard form expression f(x) = ax^n + bx^{n-1} + ..., the leading coefficient is the constant 'a'.
Question: If f(x) = -5x^2 + 20x - 3, what is the leading coefficient, and does the graph of f open upward or downward? Solution: The term with the highest degree is -5x^2. The coefficient of this term is -5. Because the leading coefficient is negative (-5 < 0), the parabola opens downward.
Mistake 1: Assuming the first term written is always the leading coefficient, even if the polynomial is not in standard descending order.
Mistake 2: Confusing the leading coefficient with the constant term (the y-intercept) when predicting the graph's behavior.
Mistake 3: Ignoring the sign of the leading coefficient when determining if a quadratic has a maximum or minimum value.
Students targeting 750+ should know that the leading coefficient, when combined with the degree of the polynomial, dictates the global end behavior; for even-degree polynomials, a positive leading coefficient means both ends point to positive infinity, while for odd-degree polynomials, it determines if the graph starts at negative infinity and ends at positive infinity.
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