What is the standard form of a quadratic polynomial in one variable?

• A. ax^2 + bx + c
• B. ax^3 + bx^2 + cx + d
• C. a x^2 + c
• D. a x + b

Answer: (A)

The main idea is that a quadratic polynomial in one variable must have an x^2 term, an x term, and a constant term, with the x^2 coefficient not equal to zero. This gives the standard form a x^2 + b x + c, where a ≠ 0. Here, a determines the parabola’s opening and width, b shifts the slope via the linear term, and c moves the graph up or down. If the x^2 term is missing or the degree is higher than 2, you’re not looking at a quadratic. If the linear term is omitted, you still wouldn’t be in the conventional standard form. So the expression with all three parts—ax^2, bx, and c—fits the standard form exactly.