Question

How do you determine if a polynomial equation is a perfect square?

a) Check if the leading coefficient is a perfect square.
b) Check if the constant term is a perfect square.
c) Factor the polynomial and see if it can be written as (ax ± b)^2.
d) Count the number of terms in the polynomial.

Answer 1

To determine if a polynomial equation is a perfect square, check if the leading coefficient and constant term are perfect squares and if the middle term is twice the product of their square roots.

Step-by-step explanation:

To determine if a polynomial equation is a perfect square, we need to look for certain characteristics. A polynomial is a perfect square if it can be expressed in the form of (ax ± b)^2, where a and b are real numbers. A perfect square trinomial is a special case of a polynomial that has three terms. Here are the steps to identify a perfect square trinomial:

1. Check if the leading coefficient is a perfect square.
2. Ensure the constant term is also a perfect square.
3. The middle term should be twice the product of the square roots of the leading coefficient and the constant term.
4. Factor the polynomial completely; a perfect square trinomial can be factored into (ax ± b)^2.

For example, x^2 + 2x + 1 is a perfect square because it can be factored as (x + 1)^2. The leading coefficient (1) and the constant term (1) are both perfect squares, and the middle term (2x) is twice the product of the square roots of 1 (i.e., 1 × 2).

Not all polynomials are perfect squares, and even if both the leading coefficient and constant term are perfect squares, it doesn't guarantee that the polynomial is a perfect square without the right middle term. For instance, x^2 + 2x + 4 is not a perfect square since 4 is not the square of the middle term's coefficient (2).

In summary, to solve for an unknown value when faced with a quadratic equation that is a perfect square, it's easier to factor it as such rather than using the quadratic formula.