03Efactorisingcubics

= Factorising Cubics =


 * Once one factor has been found, possibly by using the factor theorem.
 * The cubic can be divided by that factor to find a quadratic factor
 * That quadratic factor might then be factorised to get two other linear factors.


 * Example 1 **

... ... Factorise x 3 + 7x 2 + 7x – 15 ... ... Given that (x + 3) is a factor.


 * Solution:**

Long Division



From long division, we get that math . \qquad x^3 + 7x^2 + 7x - 15 \\. \\ . \qquad = \big( x + 3 \big) \big( x^2 + 4x - 5 \big) math

Now factorise the quadratic

math . \qquad = \big( x + 3 \big) \big( x + 5 \big) \big( x - 1 \big) math

Short Division

If we know the divisor is a factor of the polynomial, we can do the following:

Example 2

... ... Factorise (x 3 – 2x 2 – 3x + 6) given that (x – 2) is a factor


 * Solution:**


 * By examination of the above, we can conclude that the other factor must be a quadratic.


 * Further the first term must be x 2 and the last term must be –3 so we can write:

... ... x 3 – 2x 2 – 3x + 6 = (x – 2)(x 2 + ** a **x – 3) ... ... where ** a ** is an unknown


 * If we expand these brackets we get:

... ... x 3 __– 2__x 2 – 3x + 6 = x 3 __+ ** a **__x 2 – 3x __– 2__x 2 – 2** a **x + 6


 * Equate the coefficients of x 2 (underlined above), we get:

... ... –2 = ** a ** – 2

... ... so ** a ** = 0

Thus

math . \qquad x^3 - 2x^2 - 3x + 6 = \big( x - 2 \big) \big( x^2 - 3 \big) math
 * The quadratic in the second bracket can then be factorised
 * in this case using difference of two squares

Thus math . \qquad x^3 - 2x^2 - 3x + 6 = \big( x - 2 \big) \big( x - \sqrt{3} \big) \big( x + \sqrt{3} \big) math


 * Example 3 **

... ... Factorise x 3 + 7x 2 + 7x – 15 ... ... Given that (x + 3) is a factor.


 * Solution:**

math . \qquad x^3 + 7x^2 + 7x - 15 \\. \\ . \qquad = \big( x + 3 \big) \big(x^2 + ax - 5 \big) math

... ... ** Expand the brackets **

math . \qquad = x^3 + ax^2 - 5x + 3x^2 + 3ax - 15 math

... ... ** Look at the coefficients of x 2 **.

math . \qquad 7x^2 = ax^2 + 3x^2 \\. \\ . \qquad 7 = a + 3 \\. \\ . \qquad a = 4 math

So

math . \qquad x^3 + 7x^2 + 7x - 15 \\. \\ . \qquad = \big( x + 3 \big) \big(x^2 + 4x - 5 \big) \\. \\ . \qquad = \big( x + 3 \big) \big(x + 5 \big) \big( x - 1 \big) math


 * Factorising Cubics on the Classpad Calculator **


 * Your calculator can factorise cubics
 * The factor command is in the ACTION menu, TRANSFORMATION submenu.
 * Notice that factor will give the __rational__ factors (in the 2nd example)
 * To get the __surd__ factors, you need to use the rFactor command
 * rFactor is also in the ACTION menu, TRANSFORMATION submenu
 * Your calculator should be set to STANDARD and not DECIMAL

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