Video Transcript
Subtract six 𝑥 cubed minus four 𝑦 squared minus 𝑧 from the sum of nine 𝑥 cubed plus eight 𝑦 squared minus seven 𝑧 and eight 𝑥 cubed minus nine 𝑦 squared minus six 𝑧.
Before we start this question, it is worth remembering our rules when two signs are touching. If we have two positive signs, the resultant sign is also positive or an addition sign. If the signs are different, a positive and a negative sign, the resultant sign is a negative. If we have two negative signs, the resultant sign is a positive one. The word sum means add, so our first step is to add nine 𝑥 cubed plus eight 𝑦 squared minus seven 𝑧 and eight 𝑥 cubed minus nine 𝑦 squared minus six 𝑧.
We now need to collect the like terms, firstly nine 𝑥 cubed and eight 𝑥 cubed. Nine 𝑥 cubed plus eight 𝑥 cubed is equal to 17𝑥 cubed as nine plus eight equals 17. Next, we need to group or collect the 𝑦 squared terms. Here we have positive eight 𝑦 squared plus negative nine 𝑦 squared. The positive and negative sign become a negative, so we have positive eight 𝑦 squared minus nine 𝑦 squared. This is equal to negative 𝑦 squared as eight minus nine equals negative one.
Finally in this part of the question, we need to group the negative seven 𝑧 and the negative six 𝑧. Once again, the signs in a middle become a negative or subtraction sign. We have negative seven 𝑧 minus six 𝑧. Negative seven minus six is equal to negative 13. So we have negative 13𝑧. We can therefore say that nine 𝑥 cubed plus eight 𝑦 squared minus seven 𝑧 plus eight 𝑥 cubed minus nine 𝑦 squared minus six 𝑧 is equal to 17𝑥 cubed minus 𝑦 squared minus 13𝑧.
Our next step is to subtract six 𝑥 cubed minus four 𝑦 squared minus 𝑧 from this expression. We will repeat the same process as in the first part. Grouping the 𝑥 cubed terms gives us 17𝑥 cubed minus six 𝑥 cubed. This is equal to 11𝑥 cubed. Grouping the 𝑦 squared terms gives us negative 𝑦 squared minus negative four 𝑦 squared. This time, as we have two negative signs touching, this turns into a positive. So we have negative 𝑦 squared plus four 𝑦 squared. Negative one plus four is equal to three. So we have three 𝑦 squared. Finally, we need to group or collect the 𝑧 terms: negative 13𝑧 minus negative 𝑧.
Once again the signs turn into a positive: negative 13𝑧 plus 𝑧. This is equal to negative 12𝑧 as negative 13 plus one equals negative 12. This means that our final answer when we subtract six 𝑥 cubed minus four 𝑦 squared minus 𝑧 from the sum of nine 𝑥 cubed plus eight 𝑦 squared minus seven 𝑧 and eight 𝑥 cubed minus nine 𝑦 squared minus six 𝑧 is 11𝑥 cubed plus three 𝑦 squared minus 12𝑧. We could check this answer by looking at the coefficients of each of the terms; that is, the numbers in front of the 𝑥 cubed, 𝑦 squared, and the 𝑧.
The coefficients of 𝑥 cubed were nine eight and six. We needed to add nine to eight, and then subtract six. Nine plus eight is equal to 17. Subtracting six gives us 11. So the first term is correct. The coefficients of 𝑦 squared were positive eight, negative nine, and negative four. We need to add positive eight and negative nine and then subtract negative four. Using our rules when the signs are touching gives us positive eight minus nine plus four. Positive eight minus nine is negative one. Adding four to this gives us three. Therefore, the coefficient of 𝑦 squared is three.
This means that the second part of our answer is also correct. The coefficients of 𝑧 were negative seven, negative six, and negative one. We need to add negative seven to negative six and then subtract negative one. Using our rules for signs touching once again gives us negative seven minus six plus one. Negative seven minus six is equal to negative 13. Adding one to this gives us negative 12. This means that the coefficient of 𝑧 is negative 12. So our third part is also correct. Our final answer is 11𝑥 cubed plus three 𝑦 squared minus 12𝑧.
