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Difference of squares

Factoring

Japanese school year: Junior high 3

What you learn

Only the x² − a² form - good for learning to spot the pattern.

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Key points

Use this to factor the difference of two squares, a2−b2a^2 - b^2. It factors into the product of the sum and difference of the original terms, (a+b)(a−b)(a + b)(a - b).

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Use this to factor the difference of fourth powers, a4−b4a^4 - b^4. First factor it into (a2+b2)(a2−b2)(a^2 + b^2)(a^2 - b^2), then factor the a2−b2a^2 - b^2 term once more.

a4−b4=(a2+b2)(a+b)(a−b)a^4 - b^4 = (a^2 + b^2)(a + b)(a - b)

The sum of squares, a2+b2a^2 + b^2, cannot be factored further over real numbers, so leave it as it is.

Worked example
x2−25x^2 - 25
The two integers that multiply to -25 and add to 0 are 5 and -5.

Choose a set to practice.