Question Video: Finding the Set of Zeros of a Cubic Function by Factorisation | Nagwa Question Video: Finding the Set of Zeros of a Cubic Function by Factorisation | Nagwa

Question Video: Finding the Set of Zeros of a Cubic Function by Factorisation Mathematics • Third Year of Preparatory School

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Find the set of zeroes of the function 𝑓(𝑥) = 𝑥(𝑥² − 81) − 2(𝑥² − 81).

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Video Transcript

Find the set of zeros of the function 𝑓 of 𝑥 equals 𝑥 times 𝑥 squared minus 81 minus two times 𝑥 squared minus 81.

To find the zeros of a function, we set the function equal to zero, which means 𝑥 times 𝑥 squared minus 81 minus two times 𝑥 squared minus 81 is equal to zero. We can solve this equation for 𝑥 to find the zeros of 𝑓. We can see immediately that all of the terms on the left-hand side have a common factor of 𝑥 squared minus 81. So we can factorize the left-hand side with this term. This gives us 𝑥 minus two times 𝑥 squared minus 81 equals zero.

Now notice that 81 is a square number, nine squared. So, in this second term, we have a difference of two squares. When we have an expression of the form 𝑎 squared minus 𝑏 squared, we can factorize this to give 𝑎 minus 𝑏 times 𝑎 plus 𝑏. In our case, this means we can factorize 𝑥 squared minus 81 to give 𝑥 minus nine times 𝑥 plus nine. We now have a product of binomial terms, linear in 𝑥. So we cannot factorize any further. This is a product of three terms, which is equal to zero. Therefore, at least one of the terms must itself be equal to zero. Therefore, either 𝑥 minus two equals zero, 𝑥 minus nine equals zero, or 𝑥 plus nine equals zero.

We can solve the first equation by adding two to both sides to give 𝑥 equals two, the second equation by adding nine to both sides to give 𝑥 equals nine, and the third equation by subtracting nine from both sides to give 𝑥 equals negative nine. Therefore, the set of zeros of 𝑓 is negative nine, two, and nine.

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