In the expression above, is a constant. If the expression is equivalent to for all , what is the value of ?
When two algebraic expressions with the same denominator are said to be equivalent for all allowed x-values, immediately multiply both sides by the common denominator to turn the problem into a polynomial identity. Then either expand and match coefficients (fastest when only one unknown coefficient like k appears) or plug in a convenient x-value that avoids making the denominator zero to form a simple equation for the unknown. This avoids doing full long division and keeps the algebra short and controlled.
Hints
Clear the denominator
Since the two expressions have the same denominator x − 3 and are equal for all x ≠ 3, think about what happens if you multiply both sides by x − 3.
Relate the numerator to a product
After you multiply both sides by x − 3, you should get an equation where the left side is the numerator x³ − 5x² + kx − 12. What product appears on the right side?
Expand and compare coefficients
Once you expand the product on the right side, compare the resulting polynomial term by term with x³ − 5x² + kx − 12. Focus on the coefficient of x to determine k.
Desmos Guide
Graph the general rational expression
In Desmos, type
(x^3 - 5x^2 + a x - 12) / (x - 3)
Desmos will prompt you to add a slider for a; accept it so a can vary.
Graph the target quadratic
On a new line, type
x^2 - 2x + 4
This graphs the quadratic that the rational expression should match for all x ≠ 3.
Adjust the slider to find the matching coefficient
Move the slider for a until the graph of the rational expression overlaps the graph of the quadratic everywhere except at x = 3 (where the rational expression has a hole). The value of a at that point is the value of k that makes the expressions equivalent.
Step-by-step Explanation
Use the fact that the denominators are the same
We are told
for all x ≠ 3. Since x − 3 ≠ 0 for those x, we can multiply both sides by x − 3 without changing the equality:
So the given numerator must equal the product on the right.
Expand the product on the right-hand side
Now expand (x − 3)(x² − 2x + 4):
So the right-hand side becomes the polynomial x³ − 5x² + 10x − 12.
Match coefficients to solve for k
From Step 1, we have
For these polynomials to be equal for all x, the coefficients of each power of x must match. The x-terms give
so k = 10. This corresponds to answer choice C.