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下列因式分解中,正确的有____.(填序号)
①$ab - 5b = b(a - 5)$; ②$2a^{2} - ab - a = a(2a - b)$;
③$-a^{2} - ab = -a(a - b)$; ④$a^{2} + 2a - 3 = a(a + 2) - 3$.
①$ab - 5b = b(a - 5)$; ②$2a^{2} - ab - a = a(2a - b)$;
③$-a^{2} - ab = -a(a - b)$; ④$a^{2} + 2a - 3 = a(a + 2) - 3$.
答案
①
1.(教材变式)下列由左边到右边的式子变形,哪些是因式分解,哪些不是?是的打"√",不是的打"×".
(1)$12x^{2}y = 4x\cdot 3xy$; () (2)$(x + 2)(x - 2) = x^{2} - 4$; ()
(3)$x^{2} + 2x = x(x + 2)$; () (4)$2x^{2} - 3x + 1 = x(2x - 3) + 1$; ()
(5)$x + 4 = x(1 + \frac{4}{x})$; () (6)$ax + bx = x(a + b)$. ()
(1)$12x^{2}y = 4x\cdot 3xy$; () (2)$(x + 2)(x - 2) = x^{2} - 4$; ()
(3)$x^{2} + 2x = x(x + 2)$; () (4)$2x^{2} - 3x + 1 = x(2x - 3) + 1$; ()
(5)$x + 4 = x(1 + \frac{4}{x})$; () (6)$ax + bx = x(a + b)$. ()
答案
(1)× (2)× (3)√ (4)× (5)× (6)√
2.(1)多项式$5y^{3} + y$的公因式是____; (2)多项式$ab + ac - a$的公因式是____;
(3)多项式$-a^{2} + 2a$的公因式是____; (4)多项式$4x^{3} - 3x^{2} + 2x$的公因式是____.
(3)多项式$-a^{2} + 2a$的公因式是____; (4)多项式$4x^{3} - 3x^{2} + 2x$的公因式是____.
答案
(1)$y$ (2)$a$ (3)$-a$ (4)$x$
3.分解因式:
(1)(2024福建中考)$x^{2} + x = $____; (2)(2024南通中考)$ax - ay = $____;
(3)$a^{2}b - 4ab + 8b = $____; (4)$-3m^{2} + 4m = $____.
(1)(2024福建中考)$x^{2} + x = $____; (2)(2024南通中考)$ax - ay = $____;
(3)$a^{2}b - 4ab + 8b = $____; (4)$-3m^{2} + 4m = $____.
答案
(1)$x(x + 1)$ (2)$a(x - y)$ (3)$b(a^{2}-4a + 8)$ (4)$-m(3m - 4)$
4.(教材变式)分解因式:
(1)$a - a^{2}$; (2)$az + bz$; (3)$6x - 3$;
(4)$8a^{2} - 15ab$; (5)$m^{2}x + n^{2}x$; (6)$ax^{2} - 2ax - a$.
(1)$a - a^{2}$; (2)$az + bz$; (3)$6x - 3$;
(4)$8a^{2} - 15ab$; (5)$m^{2}x + n^{2}x$; (6)$ax^{2} - 2ax - a$.
答案
解：(1)原式$=a(1 - a)$；
(2)原式$=z(a + b)$；
(3)原式$=3(2x - 1)$；
(4)原式$=a(8a - 15b)$；
(5)原式$=x(m^{2}+n^{2})$；
(6)原式$=a(x^{2}-2x - 1)$。
(2)原式$=z(a + b)$；
(3)原式$=3(2x - 1)$；
(4)原式$=a(8a - 15b)$；
(5)原式$=x(m^{2}+n^{2})$；
(6)原式$=a(x^{2}-2x - 1)$。
5.(教材变式)利用因式分解计算:
(1)$53.9^{2} - 43.9×53.9$; (2)$87×17.28 + 16×17.28 - 3×17.28$.
(1)$53.9^{2} - 43.9×53.9$; (2)$87×17.28 + 16×17.28 - 3×17.28$.
答案
解：(1)原式$=53.9×(53.9 - 43.9)$
$=53.9×10$
$=539$；
(2)原式$=17.28×(87 + 16 - 3)$
$=17.28×100$
$=1728$。
$=53.9×10$
$=539$；
(2)原式$=17.28×(87 + 16 - 3)$
$=17.28×100$
$=1728$。
