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Zorluk: KolaySimplifying and Factoring Algebraic Expressions

When the expression 3x2−12x−2\frac{3x^2 - 12}{x - 2} is completely simplified, it can be written in the form ax+bax + b for all x≠2x \neq 2, where aa and bb are constants. What is the value of a+ba + b?

Cevap: 9

Cevap

The value of a+ba + b is 9.
Factoring the numerator gives 3x2−12=3(x2−4)=3(x−2)(x+2)3x^2 - 12 = 3(x^2 - 4) = 3(x - 2)(x + 2). Canceling the common factor (x−2)(x - 2) in the denominator yields 3(x+2)3(x + 2), which expands to 3x+63x + 6. Matching this expression to ax+bax + b reveals a=3a = 3 and b=6b = 6. Adding these constants gives a+b=9a + b = 9.

Adım Adım Çözüm

1
Factor the numerator of the rational expression.
3x2−12=3(x2−4)=3(x−2)(x+2)3x^2 - 12 = 3(x^2 - 4) = 3(x - 2)(x + 2)
Factoring out the common numerical factor 3 and using the difference of squares identity (u2−v2=(u−v)(u+v))(u^2 - v^2 = (u-v)(u+v)) completely factors the numerator.
2
Simplify the expression by canceling common factors.
3(x−2)(x+2)x−2=3(x+2)=3x+6\frac{3(x - 2)(x + 2)}{x - 2} = 3(x + 2) = 3x + 6
Because x≠2x \neq 2, the factor (x−2)(x - 2) is non-zero and can be canceled from both numerator and denominator.
3
Identify the values of aa and bb and compute a+ba + b.
a=3a = 3, b=6b = 6, so a+b=3+6=9a + b = 3 + 6 = 9
Comparing 3x+63x + 6 to ax+bax + b gives a=3a = 3 and b=6b = 6.

Anahtar Kavram

Factoring algebraic expressions using common monomial factors and the difference of squares to simplify rational expressions.
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Simplifying and Factoring Algebraic Expressions

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