Factor. 20v^3 + 10v^2 - 2v - 1 ______

Grouping for Factoring: Group the terms into two pairs to prepare for factoring by grouping. Group the first two terms and the last two terms separately. 20v^3 + 10v^2 - 2v - 1 = (20v^3 + 10v^2) + (-2v - 1)Factor out Common Factors: Factor out the greatest common factor from each group. From the first group, factor out 10v^2. From the second group, factor out -1. (20v^3 + 10v^2) + (-2v - 1) = 10v^2(2v + 1) - 1(2v + 1)Identify Common Binomial Factor: Check if there is a common binomial factor in both groups. Both groups have a common binomial factor of (2v + 1). 10v^2(2v + 1) - 1(2v + 1) = (2v + 1)(10v^2 - 1)Factor Difference of Squares: Recognize that 10v^2 - 1 is a difference of squares and can be factored further. 10v^2 - 1 = (sqrt(10v^2))^2 - (sqrt(1))^2 = (sqrt(10)v)^2 - 1^2 = (sqrt(10)v - 1)(sqrt(10)v + 1)Final Factored Form: Write the final factored form of the polynomial. (2v + 1)(10v^2 - 1) = (2v + 1)((sqrt(10)v - 1)(sqrt(10)v + 1))

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20v^3 + 10v^2 - 2v - 1

Grouping for Factoring: Group the terms into two pairs to prepare for factoring by grouping. Group the first two terms and the last two terms separately. $20v^3 + 10v^2 - 2v - 1 = (20v^3 + 10v^2) + (-2v - 1)$
Factor out Common Factors: Factor out the greatest common factor from each group. $\newline$ From the first group, factor out $10v^2$ . From the second group, factor out $-1$ . $\newline$ $(20v^3 + 10v^2) + (-2v - 1) = 10v^2(2v + 1) - 1(2v + 1)$
Identify Common Binomial Factor: Check if there is a common binomial factor in both groups. $\newline$ Both groups have a common binomial factor of $(2v + 1)$ . $\newline$ $10v^2(2v + 1) - 1(2v + 1) = (2v + 1)(10v^2 - 1)$
Factor Difference of Squares: Recognize that $10v^2 - 1$ is a difference of squares and can be factored further. $\newline$ $10v^2 - 1 = (\sqrt{10v^2})^2 - (\sqrt{1})^2 = (\sqrt{10}v)^2 - 1^2 = (\sqrt{10}v - 1)(\sqrt{10}v + 1)$
Final Factored Form: Write the final factored form of the polynomial. $\newline$ $(2v + 1)(10v^2 - 1) = (2v + 1)(\sqrt{10}v - 1)(\sqrt{10}v + 1)$