Name: Kaleena

Who is asking: Other
Level: Secondary

Question:
Factor this equation: 5x⁷-10x⁵+4x³-8x

Hi Kaleena,

The first feature I see in this expression is that every term has an x in it. That allows me to take out an x as a common factor. Hence

5x⁷ - 10x⁵ + 4x³ - 8x

= x(5x⁶ - 10x⁴ + 4x² - 8)

What I see next is more subtile. I am going to group the terms because I see that 10 is twice 5 and 8 is twice 4.

x(5x⁶ - 10x⁴ + 4x² - 8)

= x[(5x⁶ - 10x⁴) + (4x² - 8)] In the first two terms inside the brackets take out a common factor of 5x⁴ and in the second two terms take out a common factor of 4. Thus x[(5x⁶ - 10x⁴) + (4x² - 8)]

= x[5x⁴(x² - 2) + 4(x² - 2)]

Now the two terms inside the brackets have a common factor of x² - 2. (This is the subtile observation I made when I observed that "10 is twice 5 and 8 is twice 4".) Thus

x[5x⁴(x² - 2) + 4(x² - 2)]

= x[(5x⁴ + 4)(x² - 2)]

A significant part of factoring is "pattern recognition". You need to be able to look at an expression and recognize something familiar. It's like hearing the first bar of a song and recognizing the group even if you haven't heard that exact song before.

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