Use long division to find the quotient and remainder when f(x)= 15 x^4 - 9 x^3 - 18 x^2 + 24 x + 18 is divided by g(x)= 3 x + 1
answer is
5x^3 - 14x^2/3 - 40x/9 + 256/27
+ (230/27)/(3x + 1)
this is really something you need to do yourself because I can't show you how, I can only give you the answer
To find the quotient and remainder when f(x) is divided by g(x) using long division, follow these steps:
Step 1:
Arrange the polynomial f(x) in descending order of powers of x, filling in any missing terms with coefficients of zero. In this case, f(x) = 15x^4 - 9x^3 - 18x^2 + 24x + 18.
Step 2:
Arrange the polynomial g(x) in descending order of powers of x, filling in any missing terms with coefficients of zero. In this case, g(x) = 3x + 1.
Step 3:
Divide the leading term of f(x) (15x^4) by the leading term of g(x) (3x). The result is 5x^3.
Step 4:
Multiply g(x) by the result obtained in step 3 (5x^3): 5x^3 * (3x + 1) = 15x^4 + 5x^3.
Step 5:
Subtract the result obtained in step 4 from f(x). This can be done by subtracting each corresponding term of the two polynomials. Subtracting f(x) - (15x^4 + 5x^3) gives -14x^3 - 18x^2 + 24x + 18.
Step 6:
Repeat steps 3-5 with the new polynomial -14x^3 - 18x^2 + 24x + 18.
Step 7:
Divide the new leading term (-14x^3) by the leading term of g(x) (3x). The result is -14/3 * x^2.
Step 8:
Multiply g(x) by the result obtained in step 7 (-14/3 * x^2): -14/3 * x^2 * (3x + 1) = -14/3 * x^3 - 14/3 * x^2.
Step 9:
Subtract the result obtained in step 8 from the new polynomial: (-14x^3 - 18x^2 + 24x + 18) - (-14/3 * x^3 - 14/3 * x^2) gives -14/3 * x^2 + 24x + 18 + 14/3 * x^3.
Step 10:
Repeat steps 3-9 with the new polynomial -14/3 * x^2 + 24x + 18 + 14/3 * x^3.
Step 11:
Divide the new leading term (14/3 * x^3) by the leading term of g(x) (3x). The result is 14/9 * x^2.
Step 12:
Multiply g(x) by the result obtained in step 11 (14/9 * x^2): 14/9 * x^2 * (3x + 1) = 14/3 * x^3 + 14/9 * x^2.
Step 13:
Subtract the result obtained in step 12 from the new polynomial: (-14/3 * x^2 + 24x + 18 + 14/3 * x^3) - (14/3 * x^3 + 14/9 * x^2) gives -14/9 * x^2 + 24x + 18 + 14/3 * x^3 - 14/3 *x^3 - 14/9 * x^2.
Step 14:
Repeat steps 3-13 with the new polynomial -14/9 * x^2 + 24x + 18 + 14/3 * x^3 - 14/3 *x^3 - 14/9 * x^2.
Step 15:
The process continues until a polynomial with a degree less than g(x) is obtained. At this point, the quotient is the sum of the results obtained in steps 3, 7, 11, and so on, and the remainder is the final polynomial obtained.