Question

Han is multiplying 10x⁴ by 0.5x³ and gets 5x⁷. He says that 0.5x³ is not a polynomial because 0.5 is not an integer. What is the error in Han’s thinking? Explain your reasoning.

a. Han's multiplication is correct.

b. 0.5x^3 is not a polynomial.

c. The error is in the exponent.

d. Polynomials can only have positive integer exponents.

Answer 1

Han is incorrect in believing that 0.5x^3 is not a polynomial; polynomials can have real number coefficients, and 0.5 is a real number. His multiplication of 10x^4 by 0.5x^3 to get 5x^7 is correct.

Step-by-step explanation:

The error in Han's thinking lies in his understanding of what constitutes a polynomial. A polynomial is an expression consisting of variables (also known as indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The coefficient in a polynomial can be any real number, which means it does not have to be an integer. Therefore, 0.5x3 is indeed a polynomial because it consists of a variable raised to a non-negative integer exponent, and the coefficient (0.5) is a real number.

Han's multiplication is correct: 10x4 multiplied by 0.5x3 is 5x7. When multiplying two exponentiated quantities, we multiply the coefficients (10 * 0.5 = 5) and add the exponents (4 + 3 = 7), resulting in 5x7.