简化 (x 2 + 4x)/(2x + 8)
代数是我们研究数字和变量的数学分支。数字是具有恒定值的数字，变量是不具有恒定值的字母或符号。代数基本上用于找出未知数的值。我们可以为所有相同类型的问题建立一个通用方程。
例如：
In a right-angled triangle, we have three sides. Suppose the base is represented by ‘B, perpendicular is represented by ‘P’, and the hypotenuse is represented by ‘H’ . Then according to Pythagoras theorem,
H² = P² + B²
If two sides of a triangle are given then by putting the value in the above formula we can easily get the third side.
代数表达式
数字、变量和算术运算符的系统排列称为代数表达式。例如，如果我们必须写出数学语句“从 16 中减去的数乘以三次”，则可以写为“16 – 3x”。在这里，我们假设未知数为“x”，并根据问题中给出的信息形成数学表达式。
在 16 – 3x 中，负号将表达式分成两项。因此，根据术语的数量，表达式分为以下类型。
- 单项式：如果表达式中的项数为一个，则称为单项式。示例：6c、-9d 等
- 二项式：如果表达式中的项数为两个，则该表达式称为二项式。示例：9x-3y、9t+3u 等。
- 三项式：如果表达式中的项数为三，则该表达式称为三项式。示例：a-c+d、8e+3d-12c 等。
- 多项式：如果表达式中的项数为一项或多项，则该表达式称为多项式。
Factorization: When an expression is written in the multiplication form of two or more factors is called factorization. For example, 3x² can be written as 3×x×x. Here, 3, x is the factor of 3x².
简化 (x 2 + 4x)/(2x + 8)
解决方案：
Step 1: Factorize the numerator and denominator.
= (x² + 4x)/(2x + 8)
In the numerator, x is present in both the terms so it can be common. Similarly in the denominator, 2 is present in both terms.
= (x×x + 2×2×x)/(2×x + 2×2×2)
Take x common from numerator and 2 from denominator.
= {x(x+4)}/{2(x+4)}
Step 2: Cancel the common term of numerator and denominator.
(x+4) is present in the numerator and denominator, so it can be canceled out. Write the remaining term.
= x/2
So the simplified form of the expression (x² + 4x)/(2x + 8) is x/2.
类似问题
问题 1：如有必要，将分数的分子和分母分解。也就是说，重写为产品。然后寻找并简化。假设分母不为零。 (a² + 12a)/(4a + 48)
解决方案：
Factorize the numerator and denominator.
= (a×a + 2×2×3×a)/(2×2×a + 2×2×2×2×3)
Take out the common factor from the numerator and denominator.
= {a(a + 2×2×3)}/{2×2(a + 2×2×3)}
= {a(a + 12)}/{4(a + 12)}
Cancel the common term from the numerator and denominator.
= a/4
So the simplified form of the expression (a² + 12a)/(4a + 48) is a/4.
问题 2：如有必要，将分数的分子和分母分解。也就是说，重写为产品。然后寻找并简化。假设分母不为零。 (p³ – 6p)/(3p² -18)。
解决方案：
Factorize the numerator and denominator.
= (p×p×p – 2×3×p)/(3×p×p – 2×3×3)
Take out the common factor from the numerator and denominator.
= {p( p×p – 2×3)}/{3(p×p – 2×3)}
= {p(p²-6)}/{3(p² – 6)}
Cancel the common term from the numerator and denominator.
= p/3
So the simplified form of the expression(p³ – 6p)/(3p² -18) is p/3.