91Ó°ÊÓ

Quadratic equations are fundamental in algebra. A standard quadratic equation has the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we are solving for. This type of equation forms a parabola when graphed on a coordinate plane.
A critical tool for finding the roots of any quadratic equation, especially when it doesn’t factor easily, is the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The equation may have two, one, or no real solutions. The solutions to a quadratic equation are also known as "roots." These roots can be found by setting the quadratic expression equal to zero and solving for \( x \).
In the original exercise, the transformation of the polynomial \( f(x) = 4k^2 - 8k + 15 \) into this standard form allows us to utilize the quadratic formula to find potential solutions for \( k \). This approach showcases the flexibility of quadratic equations in solving algebraic problems.

Discriminant analysis involves determining the nature of the roots of a quadratic equation. The discriminant is represented by the expression \( b^2 - 4ac \) inside the square root of the quadratic formula. It gives insight into how many and what type of solutions a quadratic equation has.
The value of the discriminant is crucial:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root, also known as a repeated root.
• If \( b^2 - 4ac < 0 \), there are no real roots, and the solutions are complex or imaginary numbers.

In our case, the discriminant for \( 4k^2 - 8k + 15 = 0 \) is calculated as \( -176 \), which is less than zero. Therefore, there are no real values for \( k \) that satisfy this equation. Discriminant analysis thus helps us conclude that \( f(x) \) is not divisible by \( x + 2 \) for any real number \( k \).

The polynomial remainder theorem is a handy tool in determining if a polynomial \( f(x) \) is divisible by a linear polynomial \( x - c \). According to this theorem, the remainder of the division of \( f(x) \) by \( x - c \) is \( f(c) \).
So, if \( f(c) = 0 \), it implies the polynomial is perfectly divisible by \( x - c \). In simpler terms, \( c \) is a root of the polynomial.
In the context of our exercise, to check the divisibility of \( f(x) \) by \( x + 2 \), we substitute \( x = -2 \) into \( f(x) \). The result, \( f(-2) \), gives us the remainder. For divisibility, \( f(-2) \) must be zero.
The calculated expression \( f(-2) = 4k^2 - 8k + 15 \) also forms a quadratic equation. By solving it, we assess if there are values for \( k \) that result in the polynomial being divisible by \( x + 2 \). Since the discriminant is negative, no real \( k \) values exist, confirming \( f(x) \) is not divisible by \( x + 2 \). This theorem simplifies testing divisibility with potential roots.