The polynomial f(x) in expanded form is
The polynomial zeros are given as:
x = 7, x = -1 and x = -3
Rewrite as:
x - 7 = 0, x + 1 = 0 and x + 3 = 0
Multiply the zeros
f(x) = (x - 7)(x + 1)(x + 3)
Expand
Further, expand
Evaluate
Hence, the polynomial f(x) in expanded form is
We have:
This gives
Expand the numerator
Rewrite as:
Factorize
Evaluate the quotient
The slant asymptote is the quotient i.e. x - 2
Hence, the slant asymptote of the function g(x) is x - 2
<h3>The discontinuities of g(x)</h3>In (b), we have:
Set the denominator to 0
Expand
Factorize
x(x + 1) - 2(x + 1) = 0
Factor our x + 1
(x - 2)(x + 1) = 0
Solve for x
x = 2 or x = -1
2 is greater than -1.
So, the discontinuities and their types are:
- Essential discontinuity at x = 2
- Removable discontinuity at x = -1
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