<< Chapter < Page Chapter >> Page >

Find the quotient: 35 a 4 b 2 + 14 a 4 b 3 42 a 2 b 4 7 a 2 b 2 .

5 a 2 + 2 a 2 b 6 b 2

Got questions? Get instant answers now!

Find the quotient: 10 x 2 + 5 x 20 5 x .

Solution

10 x 2 + 5 x 20 5 x Separate the terms. 10 x 2 5 x + 5 x 5 x 20 5 x Simplify. 2 x + 1 + 4 x

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the quotient: 18 c 2 + 6 c 9 6 c .

3 c + 1 3 2 c

Got questions? Get instant answers now!

Find the quotient: 10 d 2 5 d 2 5 d .

2 d 1 2 5 d

Got questions? Get instant answers now!

Divide a polynomial by a binomial

To divide a polynomial by a binomial    , we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

We write the long division The long division of 875 by 25.
We divide the first two digits, 87, by 25. 25 fits into 87 three times. 3 is written above the second digit of 875 in the long division bracket.
We multiply 3 times 25 and write the product under the 87. The product of 3 and 25 is 75, which is written below the first two digits of 875 in the long division bracket.
Now we subtract 75 from 87. 87 minus 75 is 12, which is written under 75.
Then we bring down the third digit of the dividend, 5. The 5 in 875 is brought down next to the 12, making 125.
Repeat the process, dividing 25 into 125. 25 fits into 125 five times. 5 is written to the right of the 3 on top of the long division bracket. 5 times 25 is 125. 125 minus 125 is zero. There is zero remainder, so 25 fits into 125 exactly five times. 875 divided by 25 equals 35.

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

35 · 25 875

Now we will divide a trinomial    by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

Find the quotient: ( x 2 + 9 x + 20 ) ÷ ( x + 5 ) .

Solution

A trinomial, x squared plus 9 x plus 20, divided by a binomial, x plus 5.
Write it as a long division problem.
Be sure the dividend is in standard form. The long division of x squared plus 9 x plus 20 by x plus 5
Divide x 2 by x . It may help to ask yourself, "What do I need to multiply x by to get x 2 ?"
Put the answer, x , in the quotient over the x term. x fits into x squared x times. x is written above the second term of x squared plus 9 x plus 20 in the long division bracket.
Multiply x times x + 5. Line up the like terms under the dividend. The product of x and x plus 5 is x squared plus 5 x, which is written below the first two terms of x squared plus 9x plus 20 in the long division bracket.
Subtract x 2 + 5 x from x 2 + 9 x .
You may find it easier to change the signs and then add.
Then bring down the last term, 20.
The sum of x squared plus 9 x and negative x squared plus negative 5 x is 4 x, which is written underneath the negative 5 x. The third term in x squared plus 9 x plus 20 is brought down next to 4 x, making 4 x plus 20.
Divide 4 x by x . It may help to ask yourself, "What do I need to
multiply x by to get 4 x ?"
Put the answer, 4, in the quotient over the constant term. 4 x divided by x is 4. Plus 4 is written on top of the long division bracket, next to x and above the 20 in x squared plus 9 x plus 20.
Multiply 4 times x + 5. x plus 5 times 4 is 4 x plus 20, which is written under the first 4 x plus 20.
Subtract 4 x + 20 from 4 x + 20. 4 x plus 20 minus 4 x plus 20 is 0. The remainder is 0. x squared plus 9 x plus 20 divided by x plus 5 equals x plus 4.
Check:
Multiply the quotient by the divisor.
( x + 4)( x + 5)
You should get the dividend.
x 2 + 9 x + 20✓

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the quotient: ( y 2 + 10 y + 21 ) ÷ ( y + 3 ) .

y + 7

Got questions? Get instant answers now!

Find the quotient: ( m 2 + 9 m + 20 ) ÷ ( m + 4 ) .

m + 5

Got questions? Get instant answers now!

When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.

Find the quotient: ( 2 x 2 5 x 3 ) ÷ ( x 3 ) .

Solution

A trinomial, 2 x squared minus 5 x minus 3, divided by a binomial, x minus 3.
Write it as a long division problem.
Be sure the dividend is in standard form. The long division of 2 x squared minus 5 x minus 3 by x minus 3.
Divide 2 x 2 by x .
Put the answer, 2 x , in the quotient over the x term.
x fits into 2 x squared 2 x times. 2 x is written above the second term of 2 x squared minus 5 x minus 3 in the long division bracket.
Multiply 2 x times x − 3. Line up the like terms under the dividend. The product of 2 x and x minus 3 is 2 x squared minus 6 x, which is written below the first two terms of 2 x squared minus 5 x minus 3 in the long division bracket.
Subtract 2 x 2 − 6 x from 2 x 2 − 5 x .
Change the signs and then add.
Then bring down the last term.
The sum of 2 x squared minus 5 x and negative 2 x squared plus 6 x is x, which is written underneath the 6 x. The third term in 2 x squared minus 5 x minus 3 is brought down next to x, making x minus 3.
Divide x by x .
Put the answer, 1, in the quotient over the constant term.
Plus 1 is written on top of the long division bracket, next to 2 x and above the minus 3 in 2 x squared minus 5 x minus 3.
Multiply 1 times x − 3. x minus 3 times 1 is x minus 3, which is written under the first x minus 3.
Subtract x − 3 from x − 3 by changing the signs and adding. The binomial x minus 3 minus the binomial negative x plus 3 is 0. The remainder is 0. 2 x squared minus 5 x minus 3 divided by x minus 3 equals 2 x plus 1.
To check, multiply ( x − 3)(2 x + 1).
The result should be 2 x 2 − 5 x − 3.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the quotient: ( 2 x 2 3 x 20 ) ÷ ( x 4 ) .

2 x + 5

Got questions? Get instant answers now!

Find the quotient: ( 3 x 2 16 x 12 ) ÷ ( x 6 ) .

3 x + 2

Got questions? Get instant answers now!

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In [link] , we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Find the quotient: ( x 3 x 2 + x + 4 ) ÷ ( x + 1 ) .

Solution

A polynomial, x cubed minus x squared plus x plus 4, divided by another polynomial, x plus 1.
Write it as a long division problem.
Be sure the dividend is in standard form. The long division of x cubed minus x squared plus x plus 4 by x plus 1.
Divide x 3 by x .
Put the answer, x 2 , in the quotient over the x 2 term.
Multiply x 2 times x + 1. Line up the like terms under the dividend.
x fits into x squared x times. x is written above the second term of x cubed minus x squared plus x plus 4 in the long division bracket.
Subtract x 3 + x 2 from x 3 x 2 by changing the signs and adding.
Then bring down the next term.
The sum of x cubed minus x squared and negative x cubed plus negative x squared is negative 2 x squared, which is written underneath the negative x squared. The next term in x cubed minus x squared plus x plus 4 is brought down next to negative 2 x squared, making negative 2 x squared plus x.
Divide −2 x 2 by x .
Put the answer, −2 x , in the quotient over the x term.
Multiply −2 x times x + 1. Line up the like terms under the dividend.
Minus 2 x is written on top of the long division bracket, next to x squared and above the x in x cubed minus x squared plus x plus 4. Negative 2 x squared minus 2 x is written under negative 2 x squared plus x.
Subtract −2 x 2 − 2 x from −2 x 2 + x by changing the signs and adding.
Then bring down the last term.
The sum of negative 2 x squared plus x and 2 x squared plus 2 x is found to be 3 x. The last term in x cubed minus x squared plus x plus 4 is brought down, making 3 x plus 4.
Divide 3 x by x .
Put the answer, 3, in the quotient over the constant term.
Multiply 3 times x + 1. Line up the like terms under the dividend.
Plus 3 is written on top of the long division bracket, above the 4 in x cubed minus x squared plus x plus 4. 3 x plus 3 is written under 3 x plus 4.
Subtract 3 x + 3 from 3 x + 4 by changing the signs and adding.
Write the remainder as a fraction with the divisor as the denominator.
The sum of 3 x plus 4 and negative 3 x plus negative 3 is 1. Therefore, the polynomial x cubed minus x squared plus x plus 4, divided by the binomial x plus 1, equals x squared minus 2 x plus the fraction 1 over x plus 1.
To check, multiply ( x + 1 ) ( x 2 2 x + 3 + 1 x + 1 ) .
The result should be x 3 x 2 + x + 4 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask