Answer:
The factored form of 4<em>m</em>³ – 28<em>m</em>² – 120<em>m</em> is 4<em>m</em>(<em>m</em> – 10)(<em>m</em> + 3). The zeroes of the function would be <em>m</em> = 0, <em>m</em> = –3, and <em>m</em> = 10.
Step-by-step explanation:
I'll give this a shot.
4<em>m</em>³ – 28<em>m</em>² – 120<em>m</em> = 0 — The original expression
4<em>m</em>³ – 28<em>m</em>² – 120<em>m</em> — Did that 0 have any purpose? I just deleted it.
4(<em>m</em>³ – 7<em>m</em>² – 30<em>m</em>) — There's a common factor in here, 4. Let's pull that aside.
4m(<em>m²</em> – 7<em>m</em> – 30) — Actually, there's <em>two</em> common factors. The second one is <em>m</em>! Let's pull <em>that</em> out too!
To factor an expression, you have to break apart the middle term, so to speak. That's only possible if you can find two numbers whose product equals that of the outside terms and whose sum equals the middle term. Here, I'm just dealing with numbers and putting that variable aside.
–30 = 10 × –3
–30 = –10 × 3
–30 = –2 × 15
–30 = 2 × –15 — To solve for any potential factors, let's find all the numbers integers that multiply to –30
Now let's see which one adds up to –7!
15 – 2 = 13 — it's not this one
2 – 15 = –13 — nor this one
10 – 3 = 7 — we're pretty close! Let's switch that negative
3 – 10 = –7 — here we go! Here's our numbers!
4<em>m</em>[(<em>m</em>² – 10<em>m</em>) + (3<em>m</em> – 30)] — now we break apart the middle term. This is <em>all</em> multiplied by 4<em>m</em>, so that still encases everything with brackets.
4<em>m</em>[<em>m</em>(<em>m</em> – 10) + 3(<em>m</em> – 10)] — Factoring the two expressions
4<em>m</em>(<em>m</em> + 3)(<em>m</em> – 10) — simplifying to find our answer! Ta-da!
