Algebraic Division

Algebraic Division

Algebraic division is used to determine factors to expressions that have a cubic or higher term. It is virtually the same as long division. For algebraic division, a linear term (called a binomial) (x +n), where n is an integer, is used as the divisor.

Example 1

Factorise x^3 + 7x^2 + 14x + 8 by dividing the cubic by (x + 1).

Set up a long division with (x+1) being the divisor

What multiplies (x + 1) to obtain an x^3 term?

The answer is x^2: place that in the x^2 column

Multiply (x + 1) by this value of x^2 to get a value of x^3 + x^2: place this in the appropriate columns and subtract

Draw down the 14x for a new expression of 6x^2 + 14x

What should multiply (x+1) to obtain an 6x^2 term? The answer is 6x

Carry on repeating until the division is complete

This gives a quadratic x^2 + 6x + 8 as an answer

Factorise the quadratic to obtain (x + 4) and (x + 2).

 x^2 + 6x + 8 (x + 1) x^3 + 7x^2 + 14x  + 8 - x^3 + x^2 6x^2 + 14x - 6x^2 + 6x 8x + 8 - 8x +8

Answer: (x + 4)(x + 2)(x + 1)

Example 2

A cubic function x^3 - 19x + 30 crosses the x-axis at x=3. What are the values of x for the two other locations where the line crosses the x-axis?

If the curve crosses the x-axis at x=3 then one of the factors must be (x - 3). Carry out an algebraic division: note that when writing out the division, the x^2 term is shown as a placeholder.

The division yields a quadratic of x^2 + 3x -10. Factorise this to obtain (x+5) and (x - 2). the other points at which the curve crosses the x-axis are (-5, 0) and (2, 0).

 x^2 + 3x - 10 (x - 3) x^3 + 0x^2 - 19x + 30 - x^3 - 3x^2 3x^2 - 19x - 3x^2 - 9x - 10x + 30 10x - 30

Answer: x = -5 and x=2