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Rewrite the expression as a product of four linear factors:

(3x^(2)-x)^(2)-6(3x^(2)-x)+8
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}-x\right)^{2}-6\left(3 x^{2}-x\right)+8$ $\newline$ Answer:

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Identify factors

Full solution

Q. Rewrite the expression as a product of four linear factors:

\newline

\left(3 x^{2}-x\right)^{2}-6\left(3 x^{2}-x\right)+8

\newline

Answer:

Recognize Quadratic Form: Recognize the given expression as a quadratic in form The expression $(3x^{2}-x)^{2}-6(3x^{2}-x)+8$ resembles a quadratic equation in the form of $(ax)^{2} - bx + c$ , where 'ax' is $(3x^2 - x)$ . To factor it, we can treat $(3x^2 - x)$ as a single variable, say 'y'. So, let $y = (3x^2 - x)$ . The expression then becomes $y^2 - 6y + 8$ .
Factor Quadratic Expression: Factor the quadratic expression $\newline$ Now we factor $y^2 - 6y + 8$ . This factors into $(y - 2)(y - 4)$ , because $(y - 2)(y - 4) = y^2 - 4y - 2y + 8 = y^2 - 6y + 8$ .
Substitute and Expand: Substitute back $(3x^2 - x)$ for $y$ We now replace $y$ with $(3x^2 - x)$ in the factors we found. So, $(y - 2)(y - 4)$ becomes $((3x^2 - x) - 2)((3x^2 - x) - 4)$ .
Find Linear Factors: Expand the factors to find the linear factors $\newline$ We need to expand $((3x^2 - x) - 2)$ and $((3x^2 - x) - 4)$ to find the linear factors. First, we simplify the expressions: $\newline$ $(3x^2 - x - 2) = (3x^2 - 3x + 2x - 2) = 3x(x - 1) + 2(x - 1) = (3x + 2)(x - 1)$ $\newline$ $(3x^2 - x - 4) = (3x^2 - 3x + 2x - 4) = 3x(x - 1) + 2(x - 2) = (3x + 2)(x - 2)$
Write Product of Factors: Write the product of four linear factors $\newline$ The product of the four linear factors is then $(3x + 2)(x - 1)(3x + 2)(x - 2)$ . However, we notice that $(3x + 2)$ is repeated, so we can write the factors as $(3x + 2)^2(x - 1)(x - 2)$ .

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