The Perfect Square Trinomial
The perfect square trinomial is a model that allows one to determine the mathematical coefficients based on prime squares. This model’s basis is an expression obtained from two identical binary terms multiplied by each other. In order to recognize an expression of this format, one should first look at the last element of the expression (“Section 1-5: Factoring polynomials”). So, for example, in the expression x2+10x+25, one needs to determine what the number 25 decomposes into. Since there is no minus in front of 10x, 25 only squares the number 5. Consequently, the construction of the square clearly shows that the expression can only be decomposed by (x+5)2. Other examples:
- x2+24x+144=(x+12)2
- x2+12x+36=(x+6)2
- x2+20x+100=(x+10)2
Sum of Cubes
The sum of cubes is an example of an expression like a3+b3, which decomposes into a simple sum multiplied by an incomplete polynomial of a quadratic equation. The decomposition of the sum of the cubes will always produce a typical expression in which only once will the sign be changed. For example, decomposing a3+b3 into multipliers will yield: (a+b)*(a2-ab+b2). If one applies the formula correctly, one can convert the expression to a simple sum of cubes, making solving large equations much easier. For example, one should decompose the sum of the cubes 8x3+27y3 into multipliers. Note that 8x3 is (2x) 3 and one should use 2x instead of a for the formula for the sum of the cubes. Then we use the cube formula, only instead of a3, we have 8x3, and instead of b3, we have 27y3. Hence, the expression 8x3+27y3 will be converted to (2x+3y)*(4x2-6xy+9y2). Other examples:
- 63 + (4x)3= (6 + 4x)(36 – 24x + 16x2)
- (7x)3 + (3y2)3=(7x + 3y2)(49x2 – 21xy2 + 9y2)
- 64x3 + 125=(4x + 5)(16x2 – 20x + 25)
Reference
Section 1-5: Factoring polynomials. (n.d.). Paul’s Online Notes.