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Which polynomial function [tex]$f(x)$[/tex] has a leading coefficient of 1, roots [tex]-4, 2[/tex], and [tex]9[/tex] with multiplicity 1, and root [tex]-5[/tex] with multiplicity 3?

A. [tex]$f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)$[/tex]
B. [tex]$f(x) = (x-5)(x-5)(x-5)(x-4)(x+2)(x+9)$[/tex]


Sagot :

To find the polynomial function \( f(x) \) with the specified roots and multiplicities, we must consider each root and their effects on the polynomial.

1. The roots given are \( -4 \), \( 2 \), and \( 9 \) each with multiplicity 1.
2. There is also a root at \( -5 \) with multiplicity 3.

Let's build the polynomial step by step:

### Step-by-step process:
1. For the root \( -4 \) with multiplicity 1, the factor corresponding to this root is \( (x + 4) \).
2. For the root \( 2 \) with multiplicity 1, the factor corresponding to this root is \( (x - 2) \).
3. For the root \( 9 \) with multiplicity 1, the factor corresponding to this root is \( (x - 9) \).
4. For the root \( -5 \) with multiplicity 3, the factor corresponding to this root is \( (x + 5)^3 \).

When combining these factors, we get the polynomial with the correct roots and multiplicities. It will look like:
[tex]\[ f(x) = (x + 5)^3 (x + 4) (x - 2) (x - 9) \][/tex]

### Leading coefficient:
- The leading coefficient is \( 1 \), as specified in the problem. This means that we do not need to multiply the entire polynomial by any constant other than \( 1 \).

### Final Polynomial:
Combining all these factors, the polynomial that satisfies all the conditions is:
[tex]\[ f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \][/tex]

So, the correct polynomial function is:
[tex]\[ f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \][/tex]

Which matches the option:
[tex]\[ \boxed{f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)} \][/tex]
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