Question

REGULARITY Rewrite (2x⁵-7x⁴-15x³+2x²+3x+6)/2x+3 as q(x)+ r(x)/g(x) using long division. a. Identify q(x), r(x) , and g(x). b. How can you check your work using the expressions of q(x), g(x) , and r(x) ?

Answer 1

To rewrite the given expression (2x⁵-7x⁴-15x³+2x²+3x+6)/2x+3 as q(x)+ r(x)/g(x) using long division, follow these steps. To check your work, substitute the quotient polynomial (q(x)) and the divisor polynomial (g(x)) back into the original expression and add the remainder polynomial (r(x)).

Step-by-step explanation:

To rewrite the given expression (2x⁵-7x⁴-15x³+2x²+3x+6)/2x+3 as q(x)+ r(x)/g(x) using long division, follow these steps:

1. Divide the leading term of the numerator (2x⁵) by the leading term of the denominator (2x) to find the quotient term (x⁴).
2. Multiply the divisor (2x+3) by the quotient term (x⁴) and subtract the result from the numerator (2x⁵-7x⁴-15x³+2x²+3x+6).
3. Repeat steps 1 and 2 until the degree of the resulting polynomial is less than the degree of the divisor.
4. Identify the quotient polynomial (q(x)) as the sum of the quotient terms obtained in step 1.
5. Identify the remainder polynomial (r(x)) as the resulting polynomial after applying all the divisions in step 2.
6. Identify the divisor polynomial (g(x)) as the original denominator (2x+3).

To check your work, substitute the quotient polynomial (q(x)) and the divisor polynomial (g(x)) back into the original expression and add the remainder polynomial (r(x)). The result should be equal to the original expression.