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Математика159. Сократите дробь: 1) x²-x-6/x-3; 2) 2x + 10/x²+x-20; 3) 2x²+9x-18/4x²-9; 4) 36a² - 12a +1/6a² + 11a-2; 5) m² + 8m-9/m² +12m +27; 6) b³-27/5b²-16b+3; 7) 9-x²/15-2x-x²; 8) y² - 8y +12/12y - y² - 20; 9) 3x²+2x-1/7x-3x²-2. 160. Постройте график функции: 1) y = x²-7x+6/x-1; 2) y = 2x²-5x+2/x-2 - x²-9/x+3
Ответ:
1)
\[\frac{x^2-x-6}{x-3}\]
\[x^2-x-6 = (x-3)(x+2)\]
\[\frac{(x-3)(x+2)}{x-3} = x+2\]
Ответ: \( x+2 \)
2)
\[\frac{2x+10}{x^2+x-20}\]
\[2x+10 = 2(x+5)\]
\[x^2+x-20 = (x+5)(x-4)\]
\[\frac{2(x+5)}{(x+5)(x-4)} = \frac{2}{x-4}\]
Ответ: \(\frac{2}{x-4}\)
3)
\[\frac{2x^2+9x-18}{4x^2-9}\]
\[2x^2+9x-18 = (2x-3)(x+6)\]
\[4x^2-9 = (2x-3)(2x+3)\]
\[\frac{(2x-3)(x+6)}{(2x-3)(2x+3)} = \frac{x+6}{2x+3}\]
Ответ: \(\frac{x+6}{2x+3}\)
4)
\[\frac{36a^2-12a+1}{6a^2+11a-2}\]
\[36a^2-12a+1 = (6a-1)^2\]
\[6a^2+11a-2 = (6a-1)(a+2)\]
\[\frac{(6a-1)^2}{(6a-1)(a+2)} = \frac{6a-1}{a+2}\]
Ответ: \(\frac{6a-1}{a+2}\)
5)
\[\frac{m^2+8m-9}{m^2+12m+27}\]
\[m^2+8m-9 = (m+9)(m-1)\]
\[m^2+12m+27 = (m+9)(m+3)\]
\[\frac{(m+9)(m-1)}{(m+9)(m+3)} = \frac{m-1}{m+3}\]
Ответ: \(\frac{m-1}{m+3}\)
6)
\[\frac{b^3-27}{5b^2-16b+3}\]
\[b^3-27 = (b-3)(b^2+3b+9)\]
\[5b^2-16b+3 = (5b-1)(b-3)\]
\[\frac{(b-3)(b^2+3b+9)}{(5b-1)(b-3)} = \frac{b^2+3b+9}{5b-1}\]
Ответ: \(\frac{b^2+3b+9}{5b-1}\)
7)
\[\frac{9-x^2}{15-2x-x^2}\]
\[9-x^2 = (3-x)(3+x)\]
\[15-2x-x^2 = -(x^2+2x-15) = -(x+5)(x-3) = (3-x)(x+5)\]
\[\frac{(3-x)(3+x)}{(3-x)(x+5)} = \frac{3+x}{x+5}\]
Ответ: \(\frac{3+x}{x+5}\)
8)
\[\frac{y^2-8y+12}{12y-y^2-20}\]
\[y^2-8y+12 = (y-6)(y-2)\]
\[12y-y^2-20 = -(y^2-12y+20) = -(y-2)(y-10) = (2-y)(y-10)\]
\[\frac{(y-6)(y-2)}{(2-y)(y-10)} = -\frac{(y-6)}{(y-10)} = \frac{y-6}{10-y}\]
Ответ: \(\frac{y-6}{10-y}\)
9)
\[\frac{3x^2+2x-1}{7x-3x^2-2}\]
\[3x^2+2x-1 = (3x-1)(x+1)\]
\[7x-3x^2-2 = -(3x^2-7x+2) = -(3x-1)(x-2) = (1-3x)(x-2)\]
\[\frac{(3x-1)(x+1)}{(1-3x)(x-2)} = -\frac{x+1}{x-2} = \frac{x+1}{2-x}\]
Ответ: \(\frac{x+1}{2-x}\)
160. Постройте график функции:
1)
\[y = \frac{x^2-7x+6}{x-1}\]
\[x^2-7x+6 = (x-1)(x-6)\]
\[y = \frac{(x-1)(x-6)}{x-1} = x-6\]
График функции \(y=x-6\) - прямая линия, проходящая через точки \((0, -6)\) и \((6, 0)\).
2)
\[y = \frac{2x^2-5x+2}{x-2} - \frac{x^2-9}{x+3}\]
\[2x^2-5x+2 = (2x-1)(x-2)\]
\[x^2-9 = (x-3)(x+3)\]
\[y = \frac{(2x-1)(x-2)}{x-2} - \frac{(x-3)(x+3)}{x+3} = 2x-1 - (x-3) = 2x-1-x+3 = x+2\]
График функции \(y=x+2\) - прямая линия, проходящая через точки \((0, 2)\) и \((-2, 0)\).
