WEBVTT
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Find the set of zeros of the function π of π₯ equals π₯ cubed plus five π₯ squared minus nine π₯ minus 45.
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This is a polynomial function.
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In particular, itβs a cubic function.
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And weβre going to find the zeros of this function by attempting to factorise the cubic expression that we have.
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We can separate the terms of this cubic expression into two groups.
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One of which is π₯ cubed plus five π₯ squared, and the other is minus nine π₯ minus 45.
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We can notice that both groups have a factor of π₯ plus five.
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π₯ cubed plus five π₯ squared is π₯ squared times π₯ plus five, and negative nine π₯ minus 45 is equal to negative nine times π₯ plus five.
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Now that we have two things with a common factor, we can combine them to get π₯ squared minus nine times π₯ plus five.
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So we have factored π of π₯ somewhat; weβve written it as a product of two factors.
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But we notice the factor of π₯ squared minus nine is a difference of two squares and so itself can be factored.
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π₯ squared minus nine is equal to π₯ plus three times π₯ minus three.
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And so π of π₯ is equal to π₯ plus three times π₯ minus three times π₯ plus five.
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Now that we have completely factored π of π₯, we can find its set of zeros.
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We wanted to find the set of values of π₯ for which π of π₯ is equal to zero.
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Using the factored form of π of π₯, we get that π₯ plus three times π₯ minus three times π₯ plus five is equal to zero.
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We have that the product of three numbers is zero, and the only way that that can happen is if one of those numbers is zero.
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So either π₯ plus three is equal to zero, or π₯ minus three is equal to zero, or π₯ plus five is equal to zero.
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So π₯ is equal to negative three, or π₯ is equal to three, or π₯ is equal to negative five.
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In the question, weβre asked for the set of zeros.
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So we need to put these three values into a set.
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π₯ is in the set that contains negative five, three, and negative three.
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The order that we write the elements of the set doesnβt matter.
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If π₯ is a zero of the function π of π₯, then itβs either negative five or three or negative three.
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And so the set of zeros of π of π₯, which is after all what weβre looking for, is the set of negative five, three, and negative three.