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