Factorize fully five 𝑦 to the fourth plus 40𝑧 squared plus 30𝑦 squared 𝑧.
Here’s our expression. The first thing we notice is that all three coefficients are divisible by five. So we want to take out a factor of five. When we do that, we’re left with
𝑦 squared [𝑦 to the fourth]. 40 divided by five equals eight. And the variable doesn’t change. 30 divided by five equals six. And the variable doesn’t change.
Notice how our leading term is 𝑦 to the fourth. We see that we have another term that has a 𝑦-variable, a 𝑦 squared. And we wanna change the order so that 𝑦 squared term comes next. It’d look like this. Five times 𝑦 to the fourth plus six 𝑦 squared 𝑧 plus eight 𝑧 squared.
Once we do that, we can see that we can factor this expression into two smaller expressions. Our 𝑦 to the fourth is found by multiplying 𝑦 squared by 𝑦 squared. And our 𝑧 squared would be found by multiplying 𝑧 times 𝑧.
We can’t forget this coefficient of eight. We need two factors that multiply together to equal eight and add together to equal six. The factors of eight are one and eight and two and four. If we add two plus four, we get six. And that means we want to use this two and four as coefficients of our 𝑧-variable, 𝑦 squared plus two 𝑧 times 𝑦 squared plus four 𝑧. And we can’t forget to bring down the five.
At this point, it’s probably good to go back and check these two expressions to make sure we end up with what we started with, 𝑦 squared plus two 𝑧 times 𝑦 squared plus four 𝑧. 𝑦 squared times 𝑦 squared equals
four 𝑦 [𝑦 to the fourth]. 𝑦 squared times four 𝑧 equals four 𝑦 squared 𝑧. Two 𝑧 times 𝑦 squared equals two 𝑦 squared 𝑧. Now we have two 𝑦 squared 𝑧 plus four 𝑦 squared 𝑧, which equals six 𝑦 squared 𝑧, and then two 𝑧 times four 𝑧, which equals eight 𝑧 squared.
We’re multiplying all of this by five. Five times 𝑦 to the fourth equals five 𝑦 to the fourth. Five times six 𝑦 squared 𝑧 equals 30𝑦 squared 𝑧. Five times eight 𝑧 squared equals 40𝑧 squared, which is what we originally started with.
The fully factorised form is five times 𝑦 squared plus two 𝑧 times 𝑦 squared plus four 𝑧.