Rewrite the expression as a product of four linear factors: (3x^(2)-14 x)^(2)-22(3x^(2)-14 x)+85 Answer:

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Rewrite the expression as a product of four linear factors:  (3x^(2)-14 x)^(2)-22(3x^(2)-14 x)+85 Answer:

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Recognize Quadratic Form: Recognize that the given expression is a quadratic in form, where the variable part is (3x^2 - 14x). Let's denote u = 3x^2 - 14x to see the structure more clearly.Rewrite in Terms of u: Rewrite the expression in terms of u: (u)^2 - 22u + 85.Factor Quadratic Expression: Factor the quadratic expression in u: (u)^2 - 22u + 85 can be factored into (u - a)(u - b), where a and b are the roots of the quadratic equation u^2 - 22u + 85 = 0.Find Roots: Find the roots of the quadratic equation by using the factoring method or the quadratic formula. The quadratic formula is u = [-b ± sqrt(b^2 - 4ac)] / (2a), where a = 1, b = -22, and c = 85.Calculate Discriminant: Calculate the discriminant: Δ = b^2 - 4ac = (-22)^2 - 4(1)(85) = 484 - 340 = 144.Find Square Root: Find the square root of the discriminant: sqrt(Δ) = sqrt(144) = 12.Use Quadratic Formula: Find the two roots using the quadratic formula: u = [22 ± 12] / 2. This gives us two roots: u = 17 and u = 5.Rewrite Factored Form: Rewrite the factored form using the roots: (u - 17)(u - 5).Substitute back for u: Substitute back 3x^2 - 14x for u to get the factored form in terms of x: (3x^2 - 14x - 17)(3x^2 - 14x - 5).Factor First Quadratic: Notice that each quadratic factor can be further factored into linear factors because they are both quadratic expressions with real roots. We will factor each quadratic separately.Write Factored Form: Factor the first quadratic 3x^2 - 14x - 17. We look for two numbers that multiply to -3*17 = -51 and add to -14. These numbers are -17 and 3.Factor Second Quadratic: Write the factored form of the first quadratic: (3x + 3)(x - 17).Write Factored Form: Factor the second quadratic 3x^2 - 14x - 5. We look for two numbers that multiply to -3*5 = -15 and add to -14. These numbers are -15 and 1.Combine Linear Factors: Write the factored form of the second quadratic: (3x + 1)(x - 5).Combine the linear factors to express the original expression as a product of four linear factors: (3x + 3)(x - 17)(3x + 1)(x - 5).

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}-14 x\right)^{2}-22\left(3 x^{2}-14 x\right)+85$ $\newline$ Answer:

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Rewrite the expression as a product of four linear factors:\newline(3x2−14x)2−22(3x2−14x)+85 \left(3 x^{2}-14 x\right)^{2}-22\left(3 x^{2}-14 x\right)+85 (3x2−14x)2−22(3x2−14x)+85\newlineAnswer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}-14 x\right)^{2}-22\left(3 x^{2}-14 x\right)+85$ $\newline$ Answer: