Question

If g(x)=(x−5)^3 (2x−7m)^4 and x=5 is a root with multiplicity n, what is the value of n?

Answer 1

If
`\displaystyle g( x) =( x-5)^(3)( 2x-7m)^(4)` and
`\displaystyle x=5` is a root with multiplicity
`\displaystyle n`, we can determine the value of
`\displaystyle n` by evaluating
`\displaystyle g( x)` at
`\displaystyle x=5`.

Substituting
`\displaystyle x=5` into
`\displaystyle g( x)`, we have:

`\displaystyle g( 5) =( 5-5)^(3)( 2( 5)-7m)^(4)`

Simplifying this expression, we get:

`\displaystyle g( 5) =( 0)^(3)( 10-7m)^(4)`

`\displaystyle g( 5) =0\cdot ( 10-7m)^(4)`

`\displaystyle g( 5) =0`

Since
`\displaystyle g( 5) =0`, it means that
`\displaystyle x=5` is a root of
`\displaystyle g( x)`. However, we need to determine the multiplicity of this root, which refers to the number of times it appears.

In this case, the root
`\displaystyle x=5` has a multiplicity of
`\displaystyle n`. Since the function
`\displaystyle g( x)` evaluates to
`\displaystyle 0` at
`\displaystyle x=5`, it implies that the root
`\displaystyle x=5` appears
`\displaystyle n` times in the factored form of
`\displaystyle g( x)`.

Therefore, the value of
`\displaystyle n` is
`\displaystyle 3` (the multiplicity of the root
`\displaystyle x=5`).

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