1. What is the numerical coefficient of -x^5 - (-2x)^5 - (-2x^5) when simplified? 2. What is the degree of -2^3? Explain your answer. 3. Is the expression 2x^{1.5} + 3x - 18 a polynomial? Why or why not?
Answer (1)
Let's break down each part of the question.
What is the numerical coefficient of − x 5 − ( − 2 x ) 5 − ( − 2 x 5 ) when simplified?
Simplify the expression:
− x 5 is already simplified.
− ( − 2 x ) 5 = − (( − 2 ) 5 × x 5 ) = − ( − 32 x 5 ) = 32 x 5 .
− ( − 2 x 5 ) = 2 x 5 .
Combine the terms: − x 5 + 32 x 5 + 2 x 5 = ( 32 + 2 − 1 ) x 5 = 33 x 5 .
The numerical coefficient is 33 .
What is the degree of − 2 3 ? Explain your answer.
− 2 3 means − ( 2 3 ) which is − 8 .
The degree of a constant (like − 8 ) is 0 because constants do not have variable parts.
Is the expression 2 x 1.5 + 3 x − 18 a polynomial? Why or why not?
A polynomial is an expression of the form a n x n + a n − 1 x n − 1 + … + a 1 x + a 0 , where the exponents n , n − 1 , … , 1 , 0 are non-negative integers.
In 2 x 1.5 + 3 x − 18 , the term x 1.5 has an exponent of 1.5, which is not an integer.
Therefore, the expression is not a polynomial because it contains a term with a non-integer exponent.
