+249 Logarithm Question: An expression is well-defined if you can compute its value without any illegal operations. For what values of x is the expression \(\frac{\sqrt{x + 1} + \sqrt{1 - x}}{\sqrt{x}}\)well defined? Express your answer in interval notation.
Polynomial Question #1: Suppose that f is a quadratic polynomial and g is a cubic polynomial, and both f and g have a leading coefficient of 1. What is the maximum degree of the polynomial (f(x))^3 - (g(x))^2 + f(x) - 1?
Polynomail Question #2: If f(x) is a polynomial of degree 7, and g(x) is a polynomial of degree 7, then what is the product of the minimum and the maximum possible degrees of f(x) + g(x)?
+23252 For the Logarithm Question:
Assuming real numbers, square roots will be defined when the value inside the square root sign is zero or positive:
sqrt(x + 1) is defined for all values: x + 1 >= 0 ---> x >= -1
sqrt(1 - x) is defined for all values: 1 - x >= 0 ---> -x >= -1 ---> x <= 1
sqrt(x) is defined for all values: x >= 0 ---> but a denominator can't be zero ---> x > 0
Putting these restrictions together: 0 < x <= 1
Polynomial Question #1:
The degreee of a polynomial is the degree of its highest degree term.
f(x) is quadratic ---> its highest degree term is 2
g(x) is cubic ---> its highest degree terms is 3
[ f(x) ]3 has a highest degree of [ x2 ]3 = x6
---> degree is 6 (with a coefficient of 1) (but it can also have terms of degree 5, degree 4, etc.)
[ g(x) ]2 has a highest degree of [ x3 ]2 = x6
---> degree is 6 (with a coefficient of 1) (but it can also have terms of degree 5, degree 4, etc.)
Since the coefficient of the x6 term of both the [ f(x) ]3 term and the [ g(x) ]2 term is 1, these terms will cancel
under subtraction, leaving a possible x5 term.
---> The maximum possible degree of the answer is 5.
Polynomial Question #2:
If f(x) = x7 and g(x) = -x7, then f(x) + g(x) = x7 + -x7 = 0, which has a degree of 0, the minimum possible degree.
If f(x) = x7 and g(x) = x7, then f(x) + g(x) = x7 + x7 = 2x7, which has a degree of 7, the maximum possible degree.
Multiplying these two answers: 0 x 7 = 0.
+23252 For the Logarithm Question:
Assuming real numbers, square roots will be defined when the value inside the square root sign is zero or positive:
sqrt(x + 1) is defined for all values: x + 1 >= 0 ---> x >= -1
sqrt(1 - x) is defined for all values: 1 - x >= 0 ---> -x >= -1 ---> x <= 1
sqrt(x) is defined for all values: x >= 0 ---> but a denominator can't be zero ---> x > 0
Putting these restrictions together: 0 < x <= 1
Polynomial Question #1:
The degreee of a polynomial is the degree of its highest degree term.
f(x) is quadratic ---> its highest degree term is 2
g(x) is cubic ---> its highest degree terms is 3
[ f(x) ]3 has a highest degree of [ x2 ]3 = x6
---> degree is 6 (with a coefficient of 1) (but it can also have terms of degree 5, degree 4, etc.)
[ g(x) ]2 has a highest degree of [ x3 ]2 = x6
---> degree is 6 (with a coefficient of 1) (but it can also have terms of degree 5, degree 4, etc.)
Since the coefficient of the x6 term of both the [ f(x) ]3 term and the [ g(x) ]2 term is 1, these terms will cancel
under subtraction, leaving a possible x5 term.
---> The maximum possible degree of the answer is 5.
Polynomial Question #2:
If f(x) = x7 and g(x) = -x7, then f(x) + g(x) = x7 + -x7 = 0, which has a degree of 0, the minimum possible degree.
If f(x) = x7 and g(x) = x7, then f(x) + g(x) = x7 + x7 = 2x7, which has a degree of 7, the maximum possible degree.
Multiplying these two answers: 0 x 7 = 0.
