is the product of two linear functions always a quadratic function? Explain..

sure

(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd
that's just a quadratic in x

No, the product of two linear functions is not always a quadratic function. A linear function is a function of the form f(x) = mx + b, where m is the slope and b is the y-intercept. When you multiply two linear functions, you get a result that involves terms with x raised to different powers.

For example, let's consider two linear functions, f(x) = 2x + 3 and g(x) = -4x + 2. When we multiply these functions, we get:

f(x) * g(x) = (2x + 3)(-4x + 2)
= -8x^2 + 4x - 12x + 6
= -8x^2 - 8x + 6

The result is a polynomial function of degree 2, also known as a quadratic function. However, it's important to note that the product of any two linear functions won't always result in a quadratic function.

In general, when you multiply two functions, the resulting function will have terms that involve different powers of x depending on the original functions and how they interact. Therefore, the product of two linear functions can be a quadratic function, but it doesn't have to be.

To determine whether the product of two linear functions is always a quadratic function, let's first understand what constitutes a linear function and a quadratic function.

A linear function is a function that can be represented by a straight line, and its general form is given by:

f(x) = mx + b

Here, "m" represents the slope of the line, and "b" is the y-intercept.

On the other hand, a quadratic function is a function of the form:

f(x) = ax^2 + bx + c

Here, "a" represents the coefficient of the quadratic term, "b" is the coefficient of the linear term, and "c" is the constant term.

Now, let's consider two linear functions:

f(x) = m1x + b1
g(x) = m2x + b2

To find the product of these two functions, we multiply them:

h(x) = f(x) * g(x)
= (m1x + b1) * (m2x + b2)

Distributing and simplifying, we get:

h(x) = m1m2x^2 + (m1b2 + m2b1)x + (b1b2)

As you can see, the product of two linear functions results in a quadratic function. The term m1m2x^2 represents the quadratic term, (m1b2 + m2b1)x represents the linear term, and b1b2 represents the constant term.

Therefore, yes, the product of two linear functions is always a quadratic function.