What is the greatest common factor of the terms of the polynomial: $$ 6 x^{5}+14 x^{4}+10 x^{3} $$ $$ 2 x $$ $$ 2 x^{5} $$ $$ 2 x^{3} $$ $$ x^{3} $$

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UpStudy AI · Accepted Answer

Factor the expression by following steps:
- step0: Factor:
  $$6x^{5}+14x^{4}+10x^{3}$$
- step1: Rewrite the expression:
  $$2x^{3}	imes 3x^{2}+2x^{3}	imes 7x+2x^{3}	imes 5$$
- step2: Factor the expression:
  $$2x^{3}\left(3x^{2}+7x+5ight)$$
Let's find the greatest common factor (GCF) of the given terms step by step.

1. **Factor the polynomial $$6x^{5} + 14x^{4} + 10x^{3}$$**:
   - The coefficients are $$6$$, $$14$$, and $$10$$. The GCF of these coefficients is $$2$$.
   - The variable part has the lowest power of $$x$$, which is $$x^{3}$$.
   - Therefore, the GCF of the polynomial is $$2x^{3}$$.

2. **Factor the other terms**:
   - For $$2x$$, the GCF is $$2x$$.
   - For $$2x^{5}$$, the GCF is $$2x^{5}$$.
   - For $$2x^{3}$$, the GCF is $$2x^{3}$$.
   - For $$x^{3}$$, the GCF is $$x^{3}$$.

3. **List the GCFs**:
   - $$6x^{5} + 14x^{4} + 10x^{3}$$: $$2x^{3}$$
   - $$2x$$: $$2x$$
   - $$2x^{5}$$: $$2x^{5}$$
   - $$2x^{3}$$: $$2x^{3}$$
   - $$x^{3}$$: $$x^{3}$$

4. **Find the overall GCF**:
   - The GCF of all the terms is determined by taking the lowest power of each factor:
     - The coefficient $$2$$ is common in all terms.
     - The variable $$x$$ has the lowest power of $$x^{1}$$ from $$2x$$.

Thus, the greatest common factor of all the terms is $$2x$$.
The greatest common factor of all the terms is $$2x$$.

What is the greatest common factor of the terms of the polynomial: \( 6 x^{5}+14 x^{4}+10 x^{3} \) \( 2 x \) \( 2 x^{5} \) \( 2 x^{3} \) \( x^{3} \)

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