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Determine the total number of roots of each polynomial function using the factored form.

[tex]\[ f(x) = (x + 1)(x - 3)(x - 4) \][/tex]

[tex]$\square$[/tex]

Sagot :

GinnyAnsweridn
To determine the total number of roots of the polynomial function [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex], let's follow these steps:

1. Identify the Factored Form:
- The given polynomial is already in its factored form: [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex].

2. Find the Roots:
- The roots of the polynomial are the values of [tex]\( x \)[/tex] that make each factor equal to zero.
- For [tex]\( (x+1) = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ x = -1 \][/tex]
- For [tex]\( (x-3) = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3 \][/tex]
- For [tex]\( (x-4) = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ x = 4 \][/tex]

3. List the Roots:
- The roots of the polynomial are [tex]\( x = -1 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = 4 \)[/tex].

4. Determine the Total Number of Roots:
- Count the roots listed: [tex]\(-1\)[/tex], [tex]\(3\)[/tex], and [tex]\(4\)[/tex].
- There are 3 roots in total.

Therefore, the total number of roots of the polynomial function [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex] is [tex]\( 3 \)[/tex].
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