WEBVTT
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Find, by factoring, the zeros of the function π of π₯ equals π₯ squared plus two π₯ minus 35.
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First, we can copy down our function.
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And then, we wanna know what do factoring means.
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It means weβll take this π₯ squared and break it into two different sums.
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π₯ plus some number times π₯ plus some number will yield π₯ squared plus two π₯ minus 35.
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The two missing values must multiply together to equal 35.
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They must be factors of 35.
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I know that one times 35 equals 35.
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That makes one and 35 factors.
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35 is not divisible by two or three or four.
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It is divisible by five.
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Five times seven equals 35.
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That makes both five and seven factors of 35.
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35 is not divisible by six.
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And that means we found all the factors.
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But how do we decide between these two sets of factors?
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Is it π₯ plus one times π₯ plus 35 or is it π₯ plus five times π₯ plus seven?
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In the next step, weβll need to look at two things: weβll need to look at the signs and weβll also need to look at the digit that represents variable π β in our case, a two.
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The first thing we see is that weβre dealing with negative 35, minus 35.
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And that means we would need to say plus one minus 35 or minus one plus 35.
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It also means we could say plus five minus seven or minus five plus seven.
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How do we go about narrowing these choices down?
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We need values that when they are added together, they equal positive two.
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If we added negative one and positive 35, we will get 34.
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Thatβs not an option.
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Okay, so what if we change those signs?
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What if we have positive one and negative 35?
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Now, we have negative 34, still not an option.
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This means we canβt use these two factors.
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Down to five and seven, here we have negative five plus seven.
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Negative five plus seven is positive two.
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What does that mean?
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It means that saying π₯ minus five times π₯ plus seven is the same thing as saying π₯ squared plus two π₯ minus 35.
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These functions are the same written in two different formats.
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But back to the task at hand: weβre trying to find the zeros.
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We want to know when would π of π₯ be equal to zero.
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Well, we know that zero times π₯ plus seven would be equal to zero or π₯ minus five times zero would be equal to zero.
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We need to know where both of these expressions would be equal to zero: when is π₯ minus five equals zero and when is π₯ plus seven equal to zero?
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On the left, we add five to both sides of our equation.
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Negative five plus five cancels out.
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And π₯ is equal to positive five.
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On the right, we subtract seven from both sides.
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Plus seven minus seven cancels out.
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And π₯ is equal to negative seven.
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The two places where this function would be equal to zero is when π₯ is equal to negative seven or five.