Final answer:
When dividing a polynomial by a binomial with missing powers of the variable, insert terms with zero coefficients for the missing powers before performing the division. For exponent rules, multiply, add, or subtract the exponents when raising a power, multiplying, or dividing exponentiated terms, respectively.
Step-by-step explanation:
When dividing a polynomial by a binomial and there are missing powers of the variable in the polynomial, we must account for these missing powers to perform the division correctly. It is essential to insert placeholder terms with coefficients of zero for the missing powers to maintain the correct order of the polynomial. This ensures that each power of the variable is represented and allows us to apply the long division method properly or synthetic division if preferred.
In terms of exponent rules, when raising a power to another power, we multiply the exponents. For multiplication of two exponentiated terms, we add the exponents if the bases are the same. In the case of division, we subtract the exponent of the divisor from the exponent of the dividend, provided the bases are the same.
For example, when dividing 3x^5 + 5 by x - 2, we would rewrite the polynomial as 3x^5 + 0x^4 + 0x^3 + 0x^2 + 0x^1 + 5 to explicitly include the missing powers of x before proceeding with the division.
To perform the division of exponentials, subtract the exponents of the exponential terms while dividing the coefficients. For instance, (6x^4) / (3x^2) would simplify to 2x^(4-2) or 2x^2.