Final answer:
To find the quotient and remainder for the division of a polynomial by a binomial, use polynomial long division. The quotient and remainder for the division of x³-x²+11x+2 by x-4 are x² + 4x + 23 and 94, respectively.
Step-by-step explanation:
To find the quotient and remainder when dividing x³-x²+11x+2 by x-4, we use polynomial long division. This is similar to long division with numbers.
Divide the first term of the numerator (x³) by the first term of the denominator (x), which gives us x². Write this above the division bar.
Multiply the entire divisor by x² and subtract this from the polynomial under the division bar.
Bring down the next term from the polynomial, and repeat the process until all terms have been accounted for.
Upon completing the division, the term on top of the division bar is the quotient, and any leftover terms are the remainder.
Example:
x² + 4x + 3
x - 4 | x³ - x² + 11x + 2
- (x³ - 4x²)
-------------
3x² + 11x
- (3x² - 12x)
-------------
23x + 2
- (23x - 92)
-------------
94
The quotient is x² + 4x + 23 and the remainder is 94.