Examkin

Question

Difficulty: EasySimplifying 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?

Answer: 9

Answer

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.

Step-by-Step Solution

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.

Key Concept

Factoring algebraic expressions using common monomial factors and the difference of squares to simplify rational expressions.
Rate this question

Topics

Simplifying and Factoring Algebraic Expressions

More questions from this topic