The leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x2 − 5x4 + 2x, the highest power is x4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph’s ends point.
In Spanish
coeficiente principal
Also called
leading term coefficient, coefficient of the highest power, lead coefficient
Reorder by power before you read the leading coefficient, and keep its sign: the sign decides whether the graph opens up or down.
Formal definition and notation
For a nonzero polynomialp(x) = anxn + an − 1xn − 1 + ⋯ + a1x + a0 with an ≠ 0, the term anxn is the leading term and its coefficientan is the leading coefficient; n is the degree. The leading coefficient is never 0. When an = 1 the polynomial is called monic. Together with the degree, the sign of an determines the end behavior of the graph.
Symbol
Read it as
Meaning
an
a sub n
the leading coefficient of a degree-n polynomial
a in ax2 + bx + c
a
the leading coefficient of a quadratic
Key facts
Find the highest power first; the leading term is not always written first. In 7 + 2x − x3, the leading coefficient is −1.
A term with no number shown has coefficient 1 or −1: −x3 means −1x3.
Quadratic ax2 + bx + c: a > 0 opens up and a < 0 opens down. A larger |a| makes the parabola narrower.
For large x, the leading term dominates: p(x) behaves like anxn, so the sign of an sets the right-hand end of the graph (up if positive, down if negative).
Every term with a variable has a coefficient; the leading coefficient is just the one on the highest-degree term.
Quick check: 3 questions
Question 1 of 3
What is the leading coefficient of 9 − 4x + 2x3?
Show answer and explanation
Answer: C. The highest power is x3 and its coefficient is 2. 9 is the constant written first, −4 goes with x, and 3 is the degree.
Question 2 of 3
What is the leading coefficient of −x5 + 3x?
Show answer and explanation
Answer: B.−x5 = −1x5, so the leading coefficient is −1. 1 drops the sign, 5 is the degree, and 3 belongs to a lower term.
Question 3 of 3
Which parabola opens downward?
Show answer and explanation
Answer: C. In standard form y = −4x2 + 5x + 2, the leading coefficient is −4, which is negative, so it opens down. The others have positive leading coefficients 3, 1 and 0.5.
Leading Coefficient: frequently asked questions
How do you find the leading coefficient?
Find the term with the highest power of the variable, then read the number in front of it, including its sign. It helps to rewrite the polynomial in standard form, from highest power to lowest. For 5 + 2x2 − 3x4, standard form is −3x4 + 2x2 + 5, so the leading coefficient is −3.
What does the leading coefficient tell you about a graph?
Its sign controls the right end of the graph: positive means the graph rises as x grows, negative means it falls. For a quadratic, a positive leading coefficient makes the parabola open up and a negative one makes it open down. Combined with the degree, it gives the full end behavior.
Can the leading coefficient be zero?
No. If the coefficient of the highest power were 0, that term would vanish, and the next nonzero term would become the leading term. For example 0x3 + 4x2 − 1 is really 4x2 − 1, with degree 2 and leading coefficient 4.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-SSE.A.1.a
Cite this entry
Effortless Math Team. “Leading Coefficient.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/leading-coefficient/
Flashcard
TermLeading Coefficientthe number on the highest power
DefinitionThe coefficient of the highest-degree term. In 3x2 − 5x4 + 2x, it is −5. Positive opens up, negative opens down.
PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Degree of a PolynomialThe degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x3 − 2x5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x2y4 has degree 6. A nonzero constant such as 9 has degree 0.Algebra · Algebra 1, Algebra 2DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, PrecalculusPowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryConstant TermA constant term is a term in an expression or equation that has no variable, so its value never changes. In 3x2 − 5x + 7, the constant term is 7. Its sign belongs to it: in x2 + 4x − 9 it is −9. In a word problem it is usually a fixed amount, such as a one-time fee.Algebra · Grade 6, Grade 7, Algebra 1Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8ConstantA constant is a fixed value that does not change. In an expression, a constant is a number without a variable, such as 8 in x + 8 or −3 in 4a − 3. Special numbers such as π are constants too. Changing the variable changes the other terms, but never the constant.Algebra · Grade 6, Grade 7, Grade 8TermA term is a single number, a variable, or numbers and variables multiplied together, such as 7, x or −4xy2. In an expression, terms are the parts separated by plus and minus signs, so 5x2 − 3x + 8 has three terms. The sign in front of a term belongs to it.Algebra · Grade 6, Grade 7, Algebra 1Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6