Answer :

Sure! Let's write the polynomial [tex]\( P(x) = 6x^5 + 49x^4 + 158x^3 + 251x^2 + 196x + 60 \)[/tex] in its factored form.

To factor this polynomial, we look for its roots, which are the values of [tex]\( x \)[/tex] where [tex]\( P(x) = 0 \)[/tex]. Once we find these roots, we can express the polynomial as a product of linear factors.

The factored form of the polynomial [tex]\( P(x) \)[/tex] is:

[tex]\[
P(x) = (x + 1)(x + 2)^2(2x + 3)(3x + 5)
\][/tex]

Here's how it breaks down:

1. Linear Factor: [tex]\((x + 1)\)[/tex]
- This means [tex]\( x = -1 \)[/tex] is a root of the polynomial.

2. Repeated Linear Factor: [tex]\((x + 2)^2\)[/tex]
- This implies [tex]\( x = -2 \)[/tex] is a root with multiplicity 2 (it repeats twice).

3. Linear Factor: [tex]\((2x + 3)\)[/tex]
- Solving [tex]\( 2x + 3 = 0 \)[/tex] gives [tex]\( x = -\frac{3}{2} \)[/tex].

4. Linear Factor: [tex]\((3x + 5)\)[/tex]
- Solving [tex]\( 3x + 5 = 0 \)[/tex] gives [tex]\( x = -\frac{5}{3} \)[/tex].

Using these factors, the polynomial can be expressed as a product of these terms. Factoring a polynomial can simplify solving equations, analyzing behavior of graphs, and finding other polynomial properties.

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Rewritten by : Barada