QuestionAugust 26, 2025

4. Factor each expression. (a) 4x^2-25 (b) 2x^2+5x-12 (c) x^3-3x^2-4x+12 (d) x^4+27x (e) 3x^3/2-9x^1/2+6x^-1/2 (f) x^3y-4xy

4. Factor each expression. (a) 4x^2-25 (b) 2x^2+5x-12 (c) x^3-3x^2-4x+12 (d) x^4+27x (e) 3x^3/2-9x^1/2+6x^-1/2 (f) x^3y-4xy
4. Factor each expression. (a) 4x^2-25 (b) 2x^2+5x-12 (c) x^3-3x^2-4x+12 (d) x^4+27x (e) 3x^3/2-9x^1/2+6x^-1/2 (f) x^3y-4xy

Solution
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Answer

(a) (2x - 5)(2x + 5) ### (b) (2x - 3)(x + 4) ### (c) (x - 2)(x + 2)(x - 3) ### (d) x(x + 3)(x^2 - 3x + 9) ### (e) 3x^{-1/2}(x - 1)(x - 2) ### (f) xy(x - 2)(x + 2) Explanation 1. Factor the difference of squares for (a) 4x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5) 2. Factor the quadratic expression for (b) Use the AC method: 2x^2 + 5x - 12. Find two numbers that multiply to 2 \times (-12) = -24 and add to 5: they are 8 and -3. Rewrite as 2x^2 + 8x - 3x - 12, then factor by grouping: (2x^2 + 8x) + (-3x - 12) = 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4). 3. Factor by grouping for (c) x^3 - 3x^2 - 4x + 12. Group terms: (x^3 - 3x^2) + (-4x + 12) = x^2(x - 3) - 4(x - 3) = (x^2 - 4)(x - 3). Further factor x^2 - 4 = (x - 2)(x + 2). So, (x - 2)(x + 2)(x - 3). 4. Factor out the greatest common factor for (d) x^4 + 27x = x(x^3 + 27). Recognize x^3 + 27 as a sum of cubes: x^3 + 3^3 = (x + 3)(x^2 - 3x + 9). Thus, x(x + 3)(x^2 - 3x + 9). 5. Factor out the greatest common factor for (e) 3x^{3/2} - 9x^{1/2} + 6x^{-1/2}. Factor out 3x^{-1/2}: 3x^{-1/2}(x^2 - 3x + 2). Factor x^2 - 3x + 2 as (x - 1)(x - 2). So, 3x^{-1/2}(x - 1)(x - 2). 6. Factor out the greatest common factor for (f) x^3y - 4xy = xy(x^2 - 4). Recognize x^2 - 4 as a difference of squares: (x - 2)(x + 2). Thus, xy(x - 2)(x + 2).

Explanation

1. Factor the difference of squares for (a)<br /> $4x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5)$<br /><br />2. Factor the quadratic expression for (b)<br /> Use the AC method: $2x^2 + 5x - 12$. Find two numbers that multiply to $2 \times (-12) = -24$ and add to $5$: they are $8$ and $-3$. Rewrite as $2x^2 + 8x - 3x - 12$, then factor by grouping: $(2x^2 + 8x) + (-3x - 12) = 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4)$.<br /><br />3. Factor by grouping for (c)<br /> $x^3 - 3x^2 - 4x + 12$. Group terms: $(x^3 - 3x^2) + (-4x + 12) = x^2(x - 3) - 4(x - 3) = (x^2 - 4)(x - 3)$. Further factor $x^2 - 4 = (x - 2)(x + 2)$. So, $(x - 2)(x + 2)(x - 3)$.<br /><br />4. Factor out the greatest common factor for (d)<br /> $x^4 + 27x = x(x^3 + 27)$. Recognize $x^3 + 27$ as a sum of cubes: $x^3 + 3^3 = (x + 3)(x^2 - 3x + 9)$. Thus, $x(x + 3)(x^2 - 3x + 9)$.<br /><br />5. Factor out the greatest common factor for (e)<br /> $3x^{3/2} - 9x^{1/2} + 6x^{-1/2}$. Factor out $3x^{-1/2}$: $3x^{-1/2}(x^2 - 3x + 2)$. Factor $x^2 - 3x + 2$ as $(x - 1)(x - 2)$. So, $3x^{-1/2}(x - 1)(x - 2)$.<br /><br />6. Factor out the greatest common factor for (f)<br /> $x^3y - 4xy = xy(x^2 - 4)$. Recognize $x^2 - 4$ as a difference of squares: $(x - 2)(x + 2)$. Thus, $xy(x - 2)(x + 2)$.
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