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Leading Term — Definition, Examples & Properties

Leading Term

The term in a polynomial which contains the highest power of the variable. For example, 5x4 is the leading term of 5x4 – 6x3 + 4x – 12.

 

 

See also

Leading coefficient

Worked Example

Problem: Find the leading term, leading coefficient, and degree of the polynomial 3x^2 + 7x^5 - 4x^3 + 9.
Step 1: Identify each term and its exponent. The terms are 3x² (exponent 2), 7x⁵ (exponent 5), −4x³ (exponent 3), and 9 (exponent 0, since 9 = 9x⁰).
3x2,7x5,−4x3,93x^2,\quad 7x^5,\quad -4x^3,\quad 9
Step 2: Find the term with the highest exponent. The largest exponent among 2, 5, 3, and 0 is 5.
max⁡(2,5,3,0)=5\max(2, 5, 3, 0) = 5
Step 3: The term with exponent 5 is 7x⁵. This is the leading term. The number in front of it, 7, is the leading coefficient. The exponent 5 is the degree of the polynomial.
Leading term=7x5\text{Leading term} = 7x^5
Answer: The leading term is 7x⁵, the leading coefficient is 7, and the degree of the polynomial is 5.

Another Example

Problem: Find the leading term of the polynomial -2x + 8x^3 - x^6 + 4x^4.
Step 1: List each term with its exponent: −2x (exponent 1), 8x³ (exponent 3), −x⁶ (exponent 6), and 4x⁴ (exponent 4).
−2x,8x3,−x6,4x4-2x,\quad 8x^3,\quad -x^6,\quad 4x^4
Step 2: The highest exponent is 6, which belongs to the term −x⁶. Note that the coefficient here is −1, not 1.
Leading term=−x6\text{Leading term} = -x^6
Answer: The leading term is −x⁶, with a leading coefficient of −1 and degree 6.

Frequently Asked Questions

Does the leading term have to be written first in a polynomial?
No. A polynomial can be written in any order. The leading term is defined by which term has the highest exponent, not by its position. However, when a polynomial is written in standard form (descending order of exponents), the leading term does appear first.
What is the difference between the leading term and the leading coefficient?
The leading term is the entire term with the highest power, including both its coefficient and its variable part (e.g., 5x⁴). The leading coefficient is just the numerical factor in front of that term (e.g., 5). The degree is just the exponent (e.g., 4).

Leading Term vs. Leading Coefficient

The leading term is the full term with the highest power of the variable, such as −3x4-3x^4. The leading coefficient is only the numerical part of that term, which in this case is −3-3. Both come from the same term, but they refer to different pieces of information.

Why It Matters

The leading term determines a polynomial's end behavior — how the graph rises or falls as xx approaches +∞+\infty or −∞-\infty. For very large values of ∣x∣|x|, the leading term dominates all other terms, so the polynomial approximately equals its leading term. Knowing the leading term also tells you the polynomial's degree, which dictates the maximum number of turning points and roots the graph can have.

Common Mistakes

Mistake: Assuming the first term written in a polynomial is always the leading term.
Correction: The leading term is the one with the highest exponent, regardless of where it appears. Always compare exponents before deciding. For example, in 4 + 3x − 2x³, the leading term is −2x³, not 4.
Mistake: Dropping the negative sign when identifying the leading term or leading coefficient.
Correction: The sign is part of the coefficient. In −x⁶, the leading coefficient is −1, not 1. This sign affects the end behavior of the polynomial's graph.

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