Question

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unit 5b: gfc & factor by grouping - homework
show your steps (full credit)!!!
find the greatest common factor and simplify the expressions.

1. $2m^5 + 4m$
2. $-2r^3 - 2r$
3. $3x^3 + 12x^2 + 9$
4. $5x^3 - 45x^2 + 70x$

Explanation:

Problem 1: \(2m^5 + 4m\)

Step1: Identify GCF of coefficients and variables

Coefficients: GCF of \(2\) and \(4\) is \(2\).
Variables: GCF of \(m^5\) and \(m\) is \(m\) (lowest power of \(m\)).
So GCF is \(2m\).

Step2: Factor out GCF

\(2m^5 + 4m = 2m \cdot m^4 + 2m \cdot 2 = 2m(m^4 + 2)\)

Step1: Identify GCF of coefficients and variables

Coefficients: GCF of \(-2\) and \(-2\) is \(-2\) (or \(2\), but factoring out \(-2\) is fine).
Variables: GCF of \(r^3\) and \(r\) is \(r\).
So GCF is \(-2r\) (or \(2r\); let's use \(-2r\) for simplicity).

Step2: Factor out GCF

\(-2r^3 - 2r = -2r \cdot r^2 + (-2r) \cdot 1 = -2r(r^2 + 1)\) (or \(2r(-r^2 - 1)\), but \(-2r(r^2 + 1)\) is simpler).

Step1: Identify GCF of coefficients and variables

Coefficients: GCF of \(3\), \(12\), \(9\) is \(3\).
Variables: No common variable (last term has no \(x\)), so GCF is \(3\).

Step2: Factor out GCF

\(3x^3 + 12x^2 + 9 = 3 \cdot x^3 + 3 \cdot 4x^2 + 3 \cdot 3 = 3(x^3 + 4x^2 + 3)\)

Answer:

\(2m(m^4 + 2)\)

Problem 2: \(-2r^3 - 2r\)