Algebra

posted by .

Divide...
1. x^3+2x^2+x+12/x+3
2. 45a^6b^10/5a^3b^2

Note that x^3+2x^2+x+12 is zero for
x = -3, therefore

(x^3+2x^2+x+12)/(x+3) has no remainder. You can divide using long division, but this is not the preferred method if you are doing the calculation by hand. If you make an error, you'll usually have to start all over again. The following method is more transparant. We put:

x^3+2x^2+x+12 = (a x^2 + b x + c) (x+3)

Work out the right hand side:

(a x^2 + b x + c) (x+3) =

a x^3 + (3 a + b) x^2 + (c + 3 b)x + 3 c

Equate the coefficients of the powers of x on both sides.

Equating coefficient of x^3 yields:

a = 1 (1)

Equating coefficient of x^2 yields:

3 a + b = 2 (2)

Equating coefficient of x yields:

c + 3 b = 1 (3)

Equating constant term yields:

3 c = 12 (4)

Inserting (1) in (2) gives:

b = -1

(4) implies that c = 4 and we see that (3) is already satisfied. The fact that there is an extra equation that is automatically satisfied has to do with the fact that the remainder is zero. If you write down the equation:

x^3+2x^2+x+12 = (a x^2 + b x + c) (x+q)

for arbitrary q, then this can only be valid if the remainder is zero. If you then proceed as above then you can rearrange the four equations into three equations for a b and c and one that says that q^3+2q^2+q+12 = 0, i.e. that q is a zero of the polynomial and that the remainder is thus zero.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Advanced Algebra

    Divide using long division or synthetic division. (21x^3 - 7)/(3x - 1) This is what I have so far, but I am having a hard time with what to do next. 3x-1/ 7(3x^3-1)
  2. Algebra

    Divide using long division or synthetic division. (21x^3 - 7)/(3x - 1) Just wondering, how would you solve this problem using synthetic division?
  3. Algebra

    Divide using long division or synthetic division. (21x^3 - 7)/(3x - 1) Just wondering, how would you solve this problem using synthetic division?
  4. Algebra

    Divide using long division: 6x3 + 19x2 + x – 10 divided by 2x -1 Divide using synthetic division: 2x3 + 2x2 + 10 divided by x + 3 PLEASE HELP!
  5. Algebra

    (4x^3-9x+8x^2-18)/(x+2)-use synthetic division to divide. I set it up like this (pretend the parentheses are the synthetic division box). (-2) 4 8 -27 4 0 -27. For remainders, according to my book, we're supposed to divide the remainder …
  6. Algebra 2 help please :)

    What is a cubic polynomial function in standard form with zeros 1, –2, and 2?
  7. algebra 1

    Simplify using long division(please show work so I understand) 1.Divide (72-8x^2+4x^3-36x)by (x-3) 2.Divide (8b^3-6) by (2b-1)
  8. Math for Educators

    I am thinking of some #s less than 100. When I divide them by 2 there is a remainder of 1. When I divide them by 3 there is a remainder of 1. When I divide it by 4 there is a remainder of 1. However, when I divide by 5 there is no …
  9. Algebra Help!

    1.) The chief designer for a large automobile company is considering 4 different radiator grills, 2 different style of headlights, and 5 different rear fender designs. If an automobile style consists of a grill, headlight, and rear …
  10. Algebra 2 (DIVIDEPOLYNOMIALS)

    How do you divide this polynomial using long division?

More Similar Questions