What is one value that makes the fraction \(\frac{(x^{2}-25)}{(x^{3}+125)}\) undefined?
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Solution
A fraction is undefined when its denominator is equal to 0. Set the denominator equal to 0 and solve for x; x3 + 125 = 0, x3 = –125, x = –5.
The fraction \(\frac{(2x^{2}+4x)}{(4x^{3}-16x^{2}-48x)}\) is equivalent to
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Solution
Factor the numerator and denominator; 2x2 + 4x = 2x(x + 2); 4x3 – 16x2 – 48x = 4x(x2 – 4x – 12) = 4x(x – 6)(x + 2). Cancel the 2x term in the numerator with the 4x term in denominator, leaving 2 in the denominator. Cancel the (x + 2) terms in the numerator and denominator, leaving 2(x – 6) = 2x – 12 in the denominator.
The fraction \(\frac{x^{2}+6x+5}{(x^{3}-25x)}\) is equivalent to
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Solution
Factor the numerator and denominator; x2 + 6x + 5 = (x + 1)(x + 5). x3 – 25x = x(x2 – 25) = x(x – 5)(x + 5). Cancel the (x + 5) terms in the numerator and denominator, leaving (x + 1) in the numerator and x(x – 5) = (x2 – 5x) in the denominator.
The fraction \(\frac{(x^{2}+8x)}{(x^{3}-64x)}\) is equivalent to
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Solution
Factor the numerator and denominator; x2 + 8x = x(x + 8); x3 – 64x = x(x2 – 64) = x(x – 8)(x + 8). Cancel the x terms and the (x + 8) terms in the numerator and denominator, leaving 1 in the numerator and (x – 8) in the denominator.
What is a root of x(x – 1)(x + 1) = 27 – x?
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Solution
First, multiply the terms on the left side of the equation. x(x – 1) = x2 – x, (x2 – x)(x + 1) = x3 + x2 – x2 – x = x3 – x. Therefore, x3 – x = 27 – x. Add x to both sides of the equation; x3 – x + x = 27 – x + x, x3 = 27. The cube root of 27 is 3, so the root, or solution, of x(x – 1)(x + 1) = 27 – x is x = 3.
What are the factors of 2x3 + 8x2 – 192x?
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Solution
The largest constant common to each term is 2, and x is the largest common variable. Factor out 2x from every term: 2x3 + 8x2 – 192x: 2x(x2 + 4x – 96). Factor x2 + 4x – 96 into (x – 8)(x + 12). The factors of 2x3 + 8x2 – 192x are 2x(x – 8)(x + 12).
What are the factors of 64x3 – 16x?
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Solution
16 is the largest constant common to 64x3 and 16x, and x is the largest common variable. Factor out 16x from both terms:\(\frac{64x^{3}}{16x}\) = 4x2 and \(\frac{-16x}{16x}\) = –1. 64x3 – 16x = 16x(4x2 – 1). Next, factor 4x2 – 1; (2x)(2x) = 4x2, and (1)(–1) = –1. 4x2 – 1 = (2x – 1)(2x + 1), so the factors of 64x3 – 16x are 16x(2x – 1)(2x + 1).
(x – 6)(x – 3)(x – 1) =
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Solution
Begin by multiplying the first two terms: (x – 6) (x – 3) = x2 – 3x – 6x + 18 = x2 – 9x + 18. Multiply (x2 – 9x + 18) by (x – 1): (x2 – 9x + 18)(x – 1) = x3 – 9x2+ 18x – x2+ 9x – 18 = x3 – 10x2+ 27x – 18.
(x2 + 5x – 7)(x + 2) =
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Solution
Multiply each term of the trinomial by each term of the binomial: (x2)(x) = x3, (5x)(x) = 5x2, (–7)(x) = –7x, (x2)(2) = 2x2, (5x)(2) = 10x, (–7)(2) = –14. Add the products and combine like terms: x3 + 5x2 + –7x + 2x2 + 10x + –14 = x3 + 7x2 + 3x – 14.
–3x(x + 6)(x – 9) =
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Solution
Begin by multiplying the first two terms: –3x(x + 6) = –3x2 – 18x. Multiply (–3x2 – 18x) by (x – 9): (–3x2 – 18x)(x – 9) = –3x3 + 27x2 – 18x2 + 162x = –3x3 + 9x2 + 162x.