Question

Why does the leading coefficient always have to be positive?

a) Because negative leading coefficients are mathematically incorrect.

b) It ensures the correct direction of the graph when the degree is even.

c) Positive leading coefficients simplify factorization.

d) Negative leading coefficients violate the fundamental theorem of algebra.

Answer 1

The leading coefficient is positive to ensure the correct direction of the graph when the degree is even and also simplifies factorization.

Step-by-step explanation:

The leading coefficient in a polynomial function is the coefficient of the term with the highest degree. It is always a constant and affects the shape and behavior of the graph. The leading coefficient being positive ensures the correct direction of the graph when the degree is even.

When the leading coefficient is positive, the graph of the function opens upwards. For example, the graph of y = 2x^2 is a U-shape, with the vertex at the lowest point on the graph. If the leading coefficient were negative, the graph would open downwards.

Additionally, positive leading coefficients simplify factorization. For example, if we have a polynomial with a positive leading coefficient, it makes it easier to find the roots or factors of the function.