4.4.2 Factorizations

Here are the most common “obscure” factorizations asked on the Number Sense exam:

Key Formulas

Sum of Cubes

$$x^3 + y^3 = [(x + y)^2 - 3xy](x + y)$$

Difference of Cubes

$$x^3 - y^3 = [(x - y)^2 + 3xy](x - y)$$

Example

Problem: If $x + y = 5$ and $xy = 3$, find $x^3 + y^3$

Solution: $x^3 + y^3 = (5^2 - 3 \cdot 3)(5) = (25 - 9)(5) = 16 \times 5 = 80$

Additional Factorizations

Identity	Formula
Sophie Germain	$x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)$
Vieta/Newton (squares)	$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$
Complete the rectangle	$(xy + x + y + 1) = (x + 1)(y + 1)$
Complete the rectangle	$(xy - x - y + 1) = (x - 1)(y - 1)$

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Problem Set 4.4.2

Practice these problems. Type your answer and press Enter to check: