5.01 Polynomials
Determine whether each of the following is a polynomial:
2 x^{3} - 4 x^{5} + 3
\sqrt{2} x^{6} + 7 x
A\left( x \right) = 4 x^{\frac{1}{4}} + 2 x^{5} + 2
3 x^{3} + \dfrac{2}{x^{7}} - 1
\left(x^{3} + 3\right) \left(\sqrt{7} + 2 x^{ - 2 }\right)
State the definition for a polynomial in terms of x.
State whether each of the following is a multivariable polynomial:
Consider the expression - 2 a^{3}.
Is this expression a monomial?
For this expression state:
The variable
The coefficient
The degree
Consider the term 9 x^{8}.
What is the coefficient of 9 x^{8}?
What is the exponent?
For each the following expressions:
5 y^{3}
- 5 y^{9} - y
x + 7 x^{6} - x^{9}
For the following polynomials, state:
P \left( x \right) = \dfrac{x^{7}}{5} + \dfrac{x^{6}}{6} + 5
P \left( x \right) = 2 x^{7} + 2 x^{5} + 2 x + 2
P \left(x\right) = 7 \sqrt{6} - \sqrt{5} x^{5} + 5
P(x)=6 - \dfrac{7}{10} x^{3}
For the following polynomials:
2 x^{4} y^{7}
7 x^{5} y + 8 x y^{3}
5 x + 0.6
7 x^{2} + 2 x - 3
- 2^{7} x y^{4}
State the degree of the following terms:
8
- 4 y^{3}
- 4^{8} y^{6}
State the numerical coefficient of:
True or False: The degree of a polynomial in x is the largest coefficient of any of the terms of the polynomial.
For the polynomial P \left( x \right) = 8:
What is the degree?
What is the constant term?
Suppose we are given a polynomial of degree n. What is the name of the process where we find two polynomials of degree less than n that multiply together to give the original?
What operation is the opposite of factorising?
Consider the expression: 12 + 7 p^{2}. What is the coefficient of p?
Which terms are like terms with 5 in the following expression?
5 + 5 x^{2} + 13 + \frac{x^{2}}{5} + \frac{1}{5} + 2 x
From 2006 to 2016, the population of Australia, measured in millions, could be estimated by the polynomial 0.02 x^{2} + 0.06 x + 20.7, where x is the number of years since 2006.
Use the polynomial to estimate the population of Australia (in millions) in 2006. Round your answer to one decimal place.
Use the polynomial to estimate the population of Australia (in millions) in 2012. Round your answer to two decimal places.
According to the estimation given by the polynomial, how much (in millions) did the population of Australia increase from 2006 to 2012? Round your answer to two decimal places.