Definition: The leading coefficient is the number multiplying the term of highest degree when a polynomial is written in standard form, with powers in descending order.
What is the leading coefficient?
Write the polynomial from the highest power down. The coefficient of the first term is the leading coefficient. In -4x3 + 7x – 1, it is -4.
Why look at it?
For a quadratic, the sign of the leading coefficient tells whether the parabola opens up or down. Its size tells how narrow the graph is. When you factor a trinomial, a leading coefficient of 1 makes the job easier.
Example. Rewrite 6 + 2x2 – 5x4 in standard form: -5x4 + 2x2 + 6. The leading coefficient is -5, not 6 and not 2. Because the leading term is negative with an even degree, the graph falls on both ends.
What is the usual error?
People read the first term as typed, not as ordered. Always rearrange first. Also, if the highest term is just x3, its coefficient is 1, which is easy to overlook. A polynomial whose leading coefficient is 1 is called monic.