Match the polynomial with its greatest common factor. Polynomial $$ 4 x^{4}-2 x^{2}+8 x $$ $$ 2 x^{3}-6 x^{2} $$ $$ 4 x^{2}-12 x-8 $$

Question

UpStudy AI · Accepted Answer

To find the greatest common factor (GCF) of each polynomial, we will factor each polynomial step by step.

### Step 1: Factor the first polynomial $$ 4x^{4} - 2x^{2} + 8x $$

1. Identify the coefficients: $$4$$, $$-2$$, and $$8$$.
2. The GCF of the coefficients is $$2$$.
3. The lowest power of $$x$$ present in all terms is $$x$$.
4. Therefore, we can factor out $$2x$$:

$$
4x^{4} - 2x^{2} + 8x = 2x(2x^{3} - x + 4)
$$

### Step 2: Factor the second polynomial $$ 2x^{3} - 6x^{2} $$

1. Identify the coefficients: $$2$$ and $$-6$$.
2. The GCF of the coefficients is $$2$$.
3. The lowest power of $$x$$ present in both terms is $$x^{2}$$.
4. Therefore, we can factor out $$2x^{2}$$:

$$
2x^{3} - 6x^{2} = 2x^{2}(x - 3)
$$

### Step 3: Factor the third polynomial $$ 4x^{2} - 12x - 8 $$

1. Identify the coefficients: $$4$$, $$-12$$, and $$-8$$.
2. The GCF of the coefficients is $$4$$.
3. The lowest power of $$x$$ present in all terms is $$x^{0}$$ (since there are constant terms).
4. Therefore, we can factor out $$4$$:

$$
4x^{2} - 12x - 8 = 4(x^{2} - 3x - 2)
$$

### Summary of GCFs

- For $$4x^{4} - 2x^{2} + 8x$$, the GCF is $$2x$$.
- For $$2x^{3} - 6x^{2}$$, the GCF is $$2x^{2}$$.
- For $$4x^{2} - 12x - 8$$, the GCF is $$4$$.

Thus, the GCFs for the polynomials are:
1. $$4x^{4} - 2x^{2} + 8x$$ → $$2x$$
2. $$2x^{3} - 6x^{2}$$ → $$2x^{2}$$
3. $$4x^{2} - 12x - 8$$ → $$4$$
The greatest common factors (GCFs) for the polynomials are: 1. $$4x^{4} - 2x^{2} + 8x$$ → $$2x$$ 2. $$2x^{3} - 6x^{2}$$ → $$2x^{2}$$ 3. $$4x^{2} - 12x - 8$$ → $$4$$

Match the polynomial with its greatest common factor. Polynomial \( 4 x^{4}-2 x^{2}+8 x \) \( 2 x^{3}-6 x^{2} \) \( 4 x^{2}-12 x-8 \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions