Factorize fully 𝑎 to the fourth power minus 14𝑎 squared 𝑏 squared plus 48𝑏 to the fourth power.
To factorize this, we’re looking for two expressions that multiply together to equal 𝑎 to the fourth minus 14𝑎 squared 𝑏 squared plus 48𝑏 to the fourth. And the first two terms of these expressions must multiply together to equal 𝑎 to the fourth power. 𝑎 cubed times 𝑎 to the first power equals 𝑎 to the fourth power.
However, this combination is not going to work. We know this because our middle term is an 𝑎 squared term. And that means the first term in both of these expressions must be 𝑎 squared. The same thing is true for that 𝑏 to the fourth power. We see a 𝑏 squared in our middle term. And so we know that the last two values in our expressions must have a 𝑏 squared variable.
Now that we’re here, we’ll look for two values that multiply together to equal 48 and that sum together to equal negative 14. Okay, if they must multiply together to equal a positive value, positive 48, but they must add together to equal a negative value, negative 14, we know that both of our factors must be negative. Negative one times negative 48 equals positive 48. Negative two times negative 24 equals 48. Negative three times negative 16 equals 48. Negative four times negative 12 equals 48. And negative six times negative eight equals 48.
Remember, their sum needs to be negative 14. Negative six plus negative eight equals negative 14. We take the negative six and the negative eight and make them the coefficients of the 𝑏 squared. These two expressions should multiply together to equal 𝑎 to the fourth minus 14𝑎 squared 𝑏 squared plus 48𝑏 to the fourth. But let’s check to make sure.
𝑎 squared times 𝑎 squared equals 𝑎 to the fourth. 𝑎 squared times negative eight 𝑏 squared equals negative eight 𝑎 squared 𝑏 squared. Negative six 𝑏 times 𝑎 squared. We can rearrange the order of the variables to write negative six 𝑎 squared 𝑏 squared. And then negative six 𝑏 squared times negative eight 𝑏 squared equals 48𝑏 to the fourth. Our two terms in the middle are like terms. We can add the negative eight and the negative six, which gives us negative 14𝑎 squared 𝑏 squared and gives us what we started with, 𝑎 to the fourth minus 14𝑎 squared 𝑏 squared plus 48𝑏 to the fourth.
Our fully factorized form, 𝑎 squared minus six 𝑏 squared times 𝑎 squared minus eight 𝑏 squared.