# Algebra

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.

1. 👍
2. 👎
3. 👁

## Similar Questions

1. ### Chemistry

A current of 45A is pass through a solution of gold salt for 1hours 45minute. Calculate Mass of gold deposited, b) number of mole deposited c) if the same current is used find the time taken for 5.5g to be deposited

2. ### chemistry

A current of 45A is passing through a solution of gold salt for 1:45 minutes a)calculate the mass of AU deposited b )number of mass deposited c) if the same current is use find the time taken for 5.5 g of the AU deposited

Carl Sonntag wanted to compare what proceeds he would receive with a simple interest note versus a simple discount note. Both had the same terms: \$19,500 at 8% for 2 years. Use ordinary interest as needed. a. Calculate the simple

4. ### muisc

In 4/4 time, what note gets the beat? A. eighth note B. quarter note C. half note D. whole note

1. ### Chemistry

A current of 45A is passed through a solution of gold salt for 1hour 45 minutes; Calculate (a)The mass of gold deposited (b)The number of moles of gold deposits d (c)If the same current is used, find the time taken for 5.5g of

1.Write 4x4x4x4x4 as a power of 4. Answer: 4^5 2. Evaluate. 72 divide 8-9 divide 3 Answer: 0 3. Give the place value for the indicated digit 8 in the number 138,350 Answer: Thousands 4. Fencing a rectangular field that measures 37

3. ### Chemistry

A current of 45A is passed through a solution of gold salt for 1 hour 45 minutes calculate (a) the mass of gold deposited (b) the numbers of moles of gold deposited (c) if the same current is used, find the time taken for 5.5g of

4. ### geometry

Geometry B Semester Exam help please i am on 13 right now 13. The length of the hypotenuse of a 30 -60 -90 triangle is 11 what is the perimeter a 11/2 +33/2 divide 3 b 33/2+11/2 divide 3 c 11+33 divide 3 d 33+11 divide 3 please

1. ### Music

1. How many beats are in one measure of 6/4? A) 2 B) 4 C) 6*** D) 9 2. Which rhythmic value gets the beat in 4/8? A) eighth note*** B) quarter note C) half note D) whole note Am I correct?

2. ### chemistry

0.222g of a divalents metal is deposited when a current of 0.45a is passed through a solution of it salts for 25mins calculate the relative atomic mass of the metal (1faraday =96500c)

3. ### math

The initial amount is 1 million. Divide it by 4. Divide the result by 5. Divide the result by 2. Divide the result by 20. Subtract 50. Divide by 3 then by 8. Subtract 1. Divide the result by 7. Add 2. Divide by 3. Add 2. Lastly,

4. ### math

hi, i need help with solving this equation for "y" 2x+3y=-2 2x+3y=-2 Get y on one side. Subtract 2x from both sides. 2x-2x+3y=-2-2x combine terms. 3y=-2-2x Divide both sides by 3. 3y/3 = (-2-2x)/3 y = ?? how do you get -2-2x to