Factoring With Two Variables: Factor Each Polynomial 12m² N²

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Factoring With Two Variablesfactor Each Polynomial12m2 N2 8mn 1d Factoring with two variables involves expressing a polynomial as a product of simpler polynomials, typically binomials or trinomials. For the polynomial $12m^2 n^2 - 8mn + 1$, the goal is to factor it completely, if possible. If the polynomial cannot be factored further (i.e., it is prime), this will be noted. The process includes recognizing patterns, applying methods such as factoring quadratics, and considering the structure of the terms. The given polynomial is: $$12m^2 n^2 - 8mn + 1$$ This expression resembles a quadratic trinomial, but in two variables. Its structure suggests the possibility of a quadratic form in terms of $mn$. To facilitate factoring, we can treat $mn$ as a single variable, say $x$, transforming the polynomial into: $$12x^2 - 8x + 1$$ This simplifies the process of factoring because it becomes a standard quadratic in $x$. The quadratic can then be factored, if possible, using methods such as trial and error, the quadratic formula, or factoring techniques for quadratics. Applying the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 12$, $b = -8$, and $c = 1$: $$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \times 12 \times 1}}{2 \times 12}$$ $$x = \frac{8 \pm \sqrt{64 - 48}}{24}$$ $$x = \frac{8 \pm \sqrt{16}}{24}$$ $$x = \frac{8 \pm 4}{24}$$ This gives two solutions: $$x = \frac{8 + 4}{24} = \frac{12}{24} = \frac{1}{2}$$ $$x = \frac{8 - 4}{24} = \frac{4}{24} = \frac{1}{6}$$ Reverting back to the original variables: $$mn = \frac{1}{2}$$ or $$mn = \frac{1}{6}$$ Therefore, the quadratic factors as: $$12x^2 - 8x + 1 = 12(x - \frac{1}{2})(x - \frac{1}{6})$$ Expressed in terms of $mn$: $$12(mn - \frac{1}{2})(mn - \frac{1}{6})$$ Expanding back to original variables: $$12 \left( mn - \frac{1}{2} \right) \left( mn - \frac{1}{6} \right)$$ which, when parentheses are expanded and simplified, yields: \[ 12 \left( mn - \frac{1}{2} \right) \left( mn - \frac{1}

Dr. Jack HW Helper · Accepted Answer

Factoring polynomials with two variables involves recognizing patterns and applying algebraic methods to express the polynomial as a product of simpler factors. The process can be straightforward when the polynomial resembles standard forms such as quadratics or difference of squares. When faced with a polynomial such as 12m^2 n^2 - 8mn + 1, the key is to interpret the terms in a way that simplifies the factorization process. In this specific case, the polynomial's structure hints at a quadratic form in terms of the product mn. Treating mn as a single variable, say x, converts the polynomial into a quadratic, 12x^2 - 8x + 1. This allows the application of standard quadratic methods such as factoring, completing the square, or quadratic formula to find its roots. Using the quadratic formula, the solutions for x are 1/2 and 1/6, leading to factors of the form (mn - 1/2) and (mn - 1/6). Multiplying these factors by the leading coefficient 12 gives the fully factored form in terms of mn: 12(mn - 1/2)(mn - 1/6) Expressed via the variables m and n, this becomes (2mn - 1)(6mn - 1). Thus, the original polynomial factors into two binomials involving the product of the variables. This approach demonstrates how recognizing the quadratic structure simplifies the overall process, enabling the polynomial to be factored completely and visibly. In general, the process of factoring with two variables entails identifying the structure, transforming the problem when necessary, and applying algebraic methods systematically. For more complex polynomials, techniques such as grouping, substitution, and advanced factorization formulas can be employed, always aiming to express the polynomial as a product of irreducible factors or determine that it is prime. References Anton, H., & Rorres, C. (2014). Fundamentals of Mathematics. Wiley. Bittinger, M. L. (2012). Elementary and Intermediate Algebra. Pearson. Gelfand, I. M., & Shen, A. (2014). Algebra. Birkhäuser. Larson, R., & Hoste

Factoring With Two Variablesfactor Each Polynomial12m2 N2 8mn 1d

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