MATH SOLVE

3 months ago

Q:
# The table represents the start of the division of 3x^4+22x^3+37x^2-7x+10by the indicated divisor

Accepted Solution

A:

The quotient is x^3 + 4x^2 -x + 1.

Solution:

By polynomial grid division, we start by the divisor 3x + 10 placed on the column headings.

3x 10

x^3 3x^4

We know that 3x^4 must be in the top left which means that the first row entry must be x^3. So the row and column multiply to 3x^4. We use this to fill in all of the first row, multiplying x^3 by the terms of the column entries.

3x 10

x^3 3x^4 10x^3

4x^2

We now got 10x^3 though we want 22x^3. The next cubic entry must then be 12x^3 so that the overall sum is 22x^3.

3x 10

x^3 3x^4 10x^3

4x^2 12x^3

Now we have 40x^2, so the next quadratic entry must be -3x^2 so that the overall sum is 37x^2.

3x 10

x^3 3x^4 10x^3

4x^2 12x^3 40x^2

-x -3x^2 -10x

This time we have -10x, so the next linear entry must be 3x so that the overall sum is 7x.

3x 10

x^3 3x^4 10x^3

4x^2 12x^3 40x^2

-x -3x^2 -10x

1 3x 10

The bottom and final term is 10, which is our desired answer. Therefore, we can now read the quotient off the first column:

3x^4+22x^3+37x^2-7x+10 / 3x + 10 = x^3 + 4x^2 -x + 1

Solution:

By polynomial grid division, we start by the divisor 3x + 10 placed on the column headings.

3x 10

x^3 3x^4

We know that 3x^4 must be in the top left which means that the first row entry must be x^3. So the row and column multiply to 3x^4. We use this to fill in all of the first row, multiplying x^3 by the terms of the column entries.

3x 10

x^3 3x^4 10x^3

4x^2

We now got 10x^3 though we want 22x^3. The next cubic entry must then be 12x^3 so that the overall sum is 22x^3.

3x 10

x^3 3x^4 10x^3

4x^2 12x^3

Now we have 40x^2, so the next quadratic entry must be -3x^2 so that the overall sum is 37x^2.

3x 10

x^3 3x^4 10x^3

4x^2 12x^3 40x^2

-x -3x^2 -10x

This time we have -10x, so the next linear entry must be 3x so that the overall sum is 7x.

3x 10

x^3 3x^4 10x^3

4x^2 12x^3 40x^2

-x -3x^2 -10x

1 3x 10

The bottom and final term is 10, which is our desired answer. Therefore, we can now read the quotient off the first column:

3x^4+22x^3+37x^2-7x+10 / 3x + 10 = x^3 + 4x^2 -x + 1