Explain the degree of the denominator must be exactly one more than that of the numerator for a rational function to have a linear oblique asymptote? Explain

If the numerator p(x) is of degree n, and the denominator q(x) is of degree n-1, then the quotient will be a linear function, with a remainder that approaches zero as x gets large.

For example, (x^3-9x^2+3x-5)/(x^2-x+3) = x-8 + (19-8x)/(x^2-x+3)
Since the remainder has degree one less in the top than in the bottom, it will go to zero as x gets huge, so the rational function p(x)/q(x) is the line y = 19-8x, plus a vanishing remainder.

That is, the slant asymptote is y = 19-8x

let's look at an example.

y = (x^3 + 2x^2 + x + 5) / (x^2 + x + 1)

by long division:

y = x+1 + (4-x)/(x^2+x+1)

now as x ---->∞, the value of (4-x)/(x^2 + x+1) ----> 0
so we are left with y = x+1, which is the linear oblique asymptote

This clearly will happen whenever the degree of the numerator is one more than the degree of the denominator giving us a result of
the form
y = mx+b + R(x)/D(x), where R(x) will be the remainder after the division
and D(x) is the original denominator

Notice the numerator is 1 degree higher than the denominator, we want that
to happen to get a linear result, plus a fraction

In numeric long division, the remainder must always be less than the divisor or else you should increase the quotient, the same is true in algebraic
division, the remainder must have a degree lower than the denominator.
e.g. 456 ÷ 11 = 41 + 5/11 , note 5 < 11
not 456 ÷ 11 = 40 + 16/11, even though this is not incorrect, we should have taken out another 11 to get 41 , leave a remainder of 5

Hope this makes sense to you.

To explain why the degree of the denominator must be exactly one more than that of the numerator for a rational function to have a linear oblique asymptote, let's first understand what a rational function is.

A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not equal to zero. It usually has the form:

f(x) = P(x) / Q(x)

Where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

An oblique asymptote is a straight line that the graph of a rational function approaches as x approaches positive or negative infinity. It occurs when the degree of the numerator is exactly one less than the degree of the denominator.

When the degree of the numerator is exactly one less than the degree of the denominator, it means that the polynomial in the denominator grows faster than the one in the numerator as x approaches infinity or negative infinity. This creates a situation where the function approaches a linear function, which is the oblique asymptote.

If the degree of the denominator is greater than one more than the degree of the numerator, the function will approach a higher-order polynomial function as x approaches infinity or negative infinity. This means that the graph will not have a linear oblique asymptote.

On the other hand, if the degree of the numerator is greater than or equal to the degree of the denominator, the rational function will have a slant asymptote, where the graph approaches a non-horizontal line. The slant asymptote occurs when the degrees of the numerator and denominator are equal or when the degree of the numerator is one more than the degree of the denominator.

Therefore, for a rational function to have a linear oblique asymptote, the degree of the denominator must be exactly one more than that of the numerator. This ensures that the function approaches a linear function as x approaches infinity or negative infinity.

To understand why the degree of the denominator must be one more than that of the numerator for a rational function to have a linear oblique asymptote, let's first define a few terms.

A rational function is a function that can be written as the ratio of two polynomials, where the numerator and denominator are both polynomials. It is typically expressed in the form:

f(x) = (ax^n + bx^(n-1) + ... + cx + d) / (ex^(m) + fx^(m-1) + ... + gx + h),

where a, b, c, d, e, f, g, and h are constants, and n and m represent the degrees of the numerator and denominator polynomials, respectively.

Now, let's consider what happens as x approaches positive or negative infinity. In a rational function, the behavior at infinity is determined by the leading term in both the numerator and denominator.

If the degree of the numerator (n) is greater than the degree of the denominator (m), then as x becomes larger or smaller, the numerator's leading term will dominate the function. In this case, the function will either go to positive or negative infinity, or it may oscillate between positive and negative infinity.

If the degree of the denominator (m) is greater than the degree of the numerator (n), then as x becomes larger or smaller, the denominator's leading term will dominate the function. In this case, the function will approach zero as x approaches infinity or negative infinity.

However, when the degree of the numerator is exactly one less than the degree of the denominator (n = m - 1), a special situation occurs. In this case, as x approaches infinity or negative infinity, the function's behavior is defined by the ratio of the leading terms in the numerator and denominator.

The ratio of the leading terms creates a linear expression, resulting in a linear oblique asymptote. The oblique asymptote represents a line that the function approaches as x becomes very large or very small.

For example, if the rational function has a numerator of degree 2 (n = 2) and a denominator of degree 3 (m = 3), the leading terms are ax^2 and ex^3, respectively. As x approaches infinity or negative infinity, the ratio of these leading terms simplifies to (a/e)x, which is a linear expression.

In summary, the degree of the denominator must be exactly one more than that of the numerator for a rational function to have a linear oblique asymptote. This condition ensures that the ratio of the leading terms in the numerator and denominator creates a linear expression, leading to a linear asymptote.