### Video Transcript

Factorize fully 𝑧 minus seven all cubed plus 343.

In this question, we are given an algebraic expression that we want to factor fully. We can first note that there is no shared factor amongst the terms. This means that we will need to check if the expression is in a form that we know how to factor. We can note that there are two terms and the first term is a cube. We can then see that 343 is seven cubed, so the given expression is the sum of two cubes. And we can recall how to factor the sum of two cubes.

We know that 𝑎 cubed plus 𝑏 cubed is equal to 𝑎 plus 𝑏 times 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. Therefore, we can factor the expression by setting 𝑎 equal to 𝑧 minus seven and 𝑏 equal to seven. This gives us 𝑧 minus seven plus seven times 𝑧 minus seven squared minus 𝑧 minus seven times seven plus seven squared.

We now want to simplify this expression. We can start by noting that negative seven plus seven is zero, so the first factor simplifies to give us 𝑧. Next, we need to expand and simplify the terms in the second factor to see if we can factor this expression any further. We start by rewriting 𝑧 minus seven all squared as 𝑧 squared minus 14𝑧 plus 49. Next, we can distribute the factor of negative seven over the parentheses to obtain negative seven 𝑧 plus 49. Then, we evaluate seven squared to be 49. We multiply the two factors to get the following expression. We can then collect the like terms in the second factor to get the expression 𝑧 times 𝑧 squared minus 21𝑧 plus 147.

We should check to see if we can factor the quadratic any further. However, there are no numbers whose product is 147 and that add to negative 21. So we cannot factor any further. Alternatively, we could say that its discriminant is negative. Hence, we were able to factor the expression fully to obtain 𝑧 times 𝑧 squared minus 21𝑧 plus 147.