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Factorize the following completely: 2.1.1. 5x3y - 10x2y2 + 15xy3 2.1.2. x3 - 3x2 - 4x + 12

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QUESTION 2

2.1. Factorize the following completely:

2.1.1. 5x3y−10x2y2+15xy35x^3y - 10x^2y^2 + 15xy^3

  1. Find the greatest common factor (GCF) of all terms.

    The GCF of the coefficients 5,−10,155, -10, 15 is 5.
    The GCF of the xx terms x3,x2,xx^3, x^2, x is xx.
    The GCF of the yy terms y,y2,y3y, y^2, y^3 is yy.
    Thus, the GCF of the entire expression is 5xy5xy.

  2. Factor out the GCF from each term.

    5x3y−10x2y2+15xy3=5xy(x2−2xy+3y2)5x^3y - 10x^2y^2 + 15xy^3 = 5xy(x^2 - 2xy + 3y^2)

    The completely factorized expression is 5xy(x2−2xy+3y2)\boxed{5xy(x^2 - 2xy + 3y^2)}.

    2.1.2. x3−3x2−4x+12x^3 - 3x^2 - 4x + 12

  3. Group the terms.

    (x3−3x2)−(4x−12)(x^3 - 3x^2) - (4x - 12)

  4. Factor out the common factor from each group.

    x2(x−3)−4(x−3)x^2(x - 3) - 4(x - 3)

  5. Factor out the common binomial factor (x−3)(x-3).

    (x−3)(x2−4)(x - 3)(x^2 - 4)

  6. Factor the difference of squares x2−4x^2 - 4 as (x−2)(x+2)(x-2)(x+2).

    (x−3)(x−2)(x+2)(x - 3)(x - 2)(x + 2)

    The completely factorized expression is (x−3)(x−2)(x+2)\boxed{(x - 3)(x - 2)(x + 2)}.

    2.2. Simplify the following:

    2.2.1. (x2−1)x+1−(x−1)(x+1)x+2\frac{(x^2-1)}{x+1} - \frac{(x-1)(x+1)}{x+2}

  7. Factor the numerator of the first fraction using the difference of squares formula, a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b).

    (x−1)(x+1)x+1−(x−1)(x+1)x+2\frac{(x-1)(x+1)}{x+1} - \frac{(x-1)(x+1)}{x+2}

  8. Simplify the first fraction by canceling the common term (x+1)(x+1), assuming x≠−1x \neq -1.

    (x−1)−(x−1)(x+1)x+2(x-1) - \frac{(x-1)(x+1)}{x+2}

  9. Find a common denominator, which is (x+2)(x+2).

    (x−1)(x+2)x+2−(x−1)(x+1)x+2\frac{(x-1)(x+2)}{x+2} - \frac{(x-1)(x+1)}{x+2}

  10. Combine the numerators over the common denominator.

    (x−1)(x+2)−(x−1)(x+1)x+2\frac{(x-1)(x+2) - (x-1)(x+1)}{x+2}

  11. Factor out (x−1)(x-1) from the numerator.

    (x−1)[(x+2)−(x+1)]x+2\frac{(x-1)[(x+2) - (x+1)]}{x+2}

  12. Simplify the expression inside the square brackets.

    (x−1)[x+2−x−1]x+2\frac{(x-1)[x+2 - x-1]}{x+2} (x−1)(1)x+2\frac{(x-1)(1)}{x+2}

    The simplified expression is x−1x+2\boxed{\frac{x-1}{x+2}}.

    2.2.2. (x−2)(x+2x2)(x-\sqrt{2})(\sqrt{x}+2x^2)

  13. Use the FOIL method to multiply the two binomials.

    (x−2)(x+2x2)=x(x)+x(2x2)−2(x)−2(2x2)(x-\sqrt{2})(\sqrt{x}+2x^2) = x(\sqrt{x}) + x(2x^2) - \sqrt{2}(\sqrt{x}) - \sqrt{2}(2x^2)

  14. Simplify each term.

    xx=x1x1/2=x3/2x\sqrt{x} = x^{1}x^{1/2} = x^{3/2} x(2x2)=2x3x(2x^2) = 2x^3 −2x=−2x-\sqrt{2}\sqrt{x} = -\sqrt{2x} −2(2x2)=−22x2-\sqrt{2}(2x^2) = -2\sqrt{2}x^2

  15. Combine the simplified terms.

    x3/2+2x3−2x−22x2x^{3/2} + 2x^3 - \sqrt{2x} - 2\sqrt{2}x^2
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