03Aexpandingcubics

= Expanding Cubics =


 * Expanding 3 linear factors (sets of brackets containing the same pronumeral) will result in a polynomial of degree 3 (a cubic)
 * The technique is to start by expanding two of the factors using FOIL
 * Then to multiply the result by the third factor.


 * It doesn't matter which two factors you expand first,
 * so look for opportunities to reduce your workload, for example, by using the difference of two squares rule.


 * Example 1 **

... ... Expand (x + 4)(x – 2)(x + 3)


 * Solution:**

math \\ . \qquad \big( x + 4 \big) \big( x - 2 \big) \big( x + 3 \big) \qquad \{ \textit{first expand 2nd and 3rd brackets} \} \\. \\ . \qquad = \big( x + 4 \big) \big( x^2 +3x - 2x - 6 \big) \\. \\ . \qquad = \big( x + 4 \big) \big( x^2 + x - 6 \big) math


 * Now use a variation of FOIL to expand the remaining two brackets.
 * Each term in the first bracket must be multiplied by every term in the second bracket
 * Resulting in a total of 6 terms in the expansion.



math . \qquad = x^3 + 5x^2 -2x - 24 math


 * Example 2 **

... ... Expand (x + 3)(x – 3)(2x + 5)


 * Solution:**

math \\ . \qquad \big( x + 3 \big) \big( x - 3 \big) \big( 2x + 5 \big) \qquad \{ \textit{expand 1st and 2nd brackets using difference of two squares} \} \\. \\ . \qquad = \big( x^2 - 9 \big) \big( 2x + 5 \big) \qquad \quad \{ \textit{now expand using FOIL} \} \\.\\ . \qquad = 2x^3 + 5x^2 - 18x - 45 math


 * Expanding Perfect Cubes **

We have two variations of a rule for expanding a perfect cube:

... ... ** (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 **.

... ... ** (a – b) 3 = a 3 – 3a 2 b + 3ab 2 – b 3 **.


 * Notice that in the second rule, the negatives appear for the odd powers of b


 * Example 3 **

... ... Expand (x – 5) 3.


 * Solution:**

math \\ . \qquad \big(x - 5 \big)^3 \\. \\ . \qquad = x^3 - 3(x)^2(5) + 3(x)(5)^2 - (5)^3 \\. \\ . \qquad = x^3 - 15x^2 + 75x - 125 math


 * Example 4 **

... ... Expand (2x + 3) 3.


 * Solution:**

math \\ . \qquad \big( 2x + 3 \big)^3 \\. \\ . \qquad = (2x)^3 + 3(2x)^2(3) + 3(2x)(3)^2 + (3)^3 \\. \\ . \qquad = 8x^3 + 36x^2 + 54x + 27 math

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