Factor the following expression completely. x^(4)-16x^(2)+5x^(3)-80 x+4x^(2)-64 Answer:

Reorder and Combine Terms: First, we should reorder the terms of the expression by descending powers of x to make it easier to factor. x^4 + 5x^3 - 16x^2 + 4x^2 - 80x - 64 Now, combine like terms. x^4 + 5x^3 - 12x^2 - 80x - 64Group and Separate Terms: Next, we look for common factors in groups of terms. We can group the first three terms and the last two terms separately. (x^4 + 5x^3 - 12x^2) - (80x + 64)Factor by Grouping: Now, factor by grouping. For the first group, we can factor out an x^2. x^2(x^2 + 5x - 12) - (80x + 64) For the second group, we can factor out a 16. x^2(x^2 + 5x - 12) - 16(5x + 4)Factor Quadratic: We now factor the quadratic x^2 + 5x - 12. This can be factored into (x + 6)(x - 2). x^2(x + 6)(x - 2) - 16(5x + 4)Final Factorization: We notice that there is no common factor between the two groups, so we cannot factor further. Therefore, the expression is completely factored.

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Q. Factor the following expression completely.

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x^{4}-16 x^{2}+5 x^{3}-80 x+4 x^{2}-64

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Answer:

Reorder and Combine Terms: First, we should reorder the terms of the expression by descending powers of $x$ to make it easier to factor. $x^4 + 5x^3 - 16x^2 + 4x^2 - 80x - 64$ Now, combine like terms. $x^4 + 5x^3 - 12x^2 - 80x - 64$
Group and Separate Terms: Next, we look for common factors in groups of terms. We can group the first three terms and the last two terms separately. $x^4 + 5x^3 - 12x^2$ - $80x + 64$
Factor by Grouping: Now, factor by grouping. For the first group, we can factor out an $x^2$ . $\newline$ $x^2(x^2 + 5x - 12) - (80x + 64)$ $\newline$ For the second group, we can factor out a $16$ . $\newline$ $x^2(x^2 + 5x - 12) - 16(5x + 4)$
Factor Quadratic: We now factor the quadratic $x^2 + 5x - 12$ . This can be factored into $(x + 6)(x - 2)$ . $\newline$ $x^2(x + 6)(x - 2) - 16(5x + 4)$
Final Factorization: We notice that there is no common factor between the two groups, so we cannot factor further. Therefore, the expression is completely factored.