Question

If S(x) = 5² - 5r - 10, what is the quotient and remainder when f(x) = x² + 7x - 5 is divided by x - 2? What is f(2)? If a polynomial f(x) is divided by x - k, what is the remainder?

Answer 1

The quotient of the division of f(x) by x - 2 is x + 9, and the remainder is -23. We also find that f(2) is 13, which is consistent with the Remainder Theorem that states the remainder is f(k) when f(x) is divided by x - k.

Step-by-step explanation:

To find the quotient and remainder when f(x) = x² + 7x - 5 is divided by x - 2, we need to perform polynomial long division or synthetic division. After dividing, we can find f(2) by simply plugging in 2 into the original polynomial function. According to the Remainder Theorem, the remainder when a polynomial f(x) is divided by x - k is just f(k).

To find the quotient and remainder for this specific case, we divide f(x) by x - 2:

• x goes into x² x times, so we multiply x - 2 by x and subtract from f(x).
• Next, x goes into the resulting polynomial 9x - 5 a total of 9 times.
• Multiplying x - 2 by 9 and subtracting again gives us the remainder.

Thus, the quotient will be x + 9, and the remainder will be -23. Now, to find f(2), we can plug in 2 into f(x) to get 2² + 7(2) - 5, which equals 4 + 14 - 5, giving us 13. Notably, this is consistent with the Remainder Theorem since the remainder when dividing by x - 2 is f(2), and both methods give us the same result, 13.