Question

28) \( 16 x y-3 x-6 x^{2}+8 y \)

Ask by Higgins Daniel. in the United States
Mar 12,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The expression \(16xy - 3x - 6x^{2} + 8y\) can be factored as \((2x + 1)(8y - 3x)\).

Solution

1. Rewrite the expression by grouping similar terms: \[ 16xy - 3x - 6x^2 + 8y = 16xy + 8y -6x^2 - 3x. \] 2. Group the terms: \[ (16xy + 8y) + (-6x^2 - 3x). \] 3. Factor out the greatest common factor from each group: - From \(16xy+8y\), factor out \(8y\): \[ 16xy+8y = 8y(2x+1). \] - From \(-6x^2-3x\), factor out \(-3x\): \[ -6x^2-3x = -3x(2x+1). \] 4. Notice that both groups now have a common factor \((2x+1)\). Factor \((2x+1)\) out: \[ 8y(2x+1)-3x(2x+1) = (2x+1)(8y-3x). \] 5. The final factored form is: \[ (2x+1)(8y-3x). \]

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The Deep Dive

To factor the expression \( 16xy - 3x - 6x^{2} + 8y \), you can rearrange it into a more manageable form: \( -6x^{2} + 16xy + 8y - 3x \). Grouping the terms will help. Combining like terms gives you \(-6x^{2} + (-3x + 16xy + 8y)\). Now, factor out common factors in groups, which can lead to discovering potential binomials. Keep in mind that factorization often involves trial and error, so don't hesitate to play around with different combinations! For a little historical twist, did you know that factoring polynomials has been around since ancient civilizations? The Babylonians and Greeks made significant contributions to algebra and geometric methods, paving the way for modern polynomial algebra. It's fascinating how these ancient techniques still influence current mathematical practices!

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