To factor the polynomial [tex]\( 98 x^3 - 28 x^2 + 2 x \)[/tex], we follow these steps:
1. Identify the Common Factor:
First, we notice that each term in the polynomial [tex]\( 98 x^3 - 28 x^2 + 2 x \)[/tex] has a common factor. We can factor out the greatest common factor (GCF) of the coefficients, which in this case is 2. Additionally, each term contains the variable [tex]\( x \)[/tex]. Thus, we can factor out [tex]\( 2x \)[/tex].
Let's factor out [tex]\( 2x \)[/tex]:
[tex]\[
98 x^3 - 28 x^2 + 2 x = 2x (49 x^2 - 14 x + 1)
\][/tex]
2. Factor the Remaining Polynomial:
Next, we focus on factoring the quadratic term within the parentheses: [tex]\( 49 x^2 - 14 x + 1 \)[/tex]. To factor this, we look for two binomials [tex]\((ax + b)(cx + d)\)[/tex] that multiply to give [tex]\( 49 x^2 - 14 x + 1 \)[/tex].
Notice that [tex]\( 49 x^2 - 14 x + 1 \)[/tex] can be factored as [tex]\( (7x - 1)^2 \)[/tex]. This is because:
[tex]\[
(7x - 1)(7x - 1) = 7x \cdot 7x + 7x \cdot (-1) + (-1) \cdot 7x + (-1) \cdot (-1) = 49x^2 - 7x - 7x + 1 = 49x^2 - 14x + 1
\][/tex]
Therefore, [tex]\( 49 x^2 - 14 x + 1 = (7 x - 1)^2 \)[/tex].
3. Combine the Factors:
Now that we have factored the quadratic term, we can combine it with the GCF we factored out initially:
[tex]\[
98 x^3 - 28 x^2 + 2 x = 2x (49 x^2 - 14 x + 1) = 2x (7 x - 1)^2
\][/tex]
Thus, our final factored form of the polynomial [tex]\( 98 x^3 - 28 x^2 + 2 x \)[/tex] is:
[tex]\[
2 x (7 x - 1)^2
\][/tex]
So, the correct answer is:
[tex]\[
2 x (7 x - 1)^2
\][/tex]
