The expression
is equivalent to for all , where , , and are real numbers. What is the value of ?
When a rational expression is said to be equivalent to a polynomial for all x except where the denominator is zero, immediately rewrite the numerator as (denominator) × (polynomial). Expand this product, then match coefficients of corresponding powers of x to form simple linear equations for the unknown coefficients. Solve only the equations you actually need (here, from the x² and constant terms) and then compute whatever combination is asked for, such as b − c, without doing extra algebra like full long division unless necessary.
Hints
Connect the fraction and the quadratic
If a fraction of polynomials is equal to a polynomial for all x except one value, what must be true about how the numerator relates to the denominator and that polynomial?
Write an equation for the numerator
Try rewriting the numerator as (x − 3)(x² + bx + c). Then expand this product so you can compare it to x³ − 4x² + kx + 6.
Use coefficient matching
After you expand (x − 3)(x² + bx + c), line up like terms with x³ − 4x² + kx + 6. What equations do you get from the x² term and the constant term? Use those to solve for b and c, then find b − c.
Desmos Guide
Enter the original rational expression
In one line, type the numerator and denominator as a function, for example f(x) = (x^3 - 4x^2 + k*x + 6)/(x - 3), replacing k with the value you found from coefficient matching.
Enter the quadratic expression
In a new line, type g(x) = x^2 + b*x + c, using the values of and you solved for.
Compare the two functions
Look at the graphs of f(x) and g(x). If your work is correct, the two graphs should lie on top of each other for all x except at x = 3, where f(x) has a hole. You can also add h(x) = f(x) - g(x) and check that, for several x-values not equal to 3, h(x) evaluates to 0, confirming your coefficients and your value of are consistent.
Step-by-step Explanation
Relate the numerator to the divisor and the quadratic
Because
is equal to for all , the division has no remainder. That means the numerator must equal the divisor times the quotient:
Expand the product (x − 3)(x² + bx + c)
Expand the right-hand side:
Combine like terms:
So we must have
Match coefficients to find b and c
Now match coefficients of the same powers of on both sides.
For :
For the constant term:
(If you want, you can also match the -coefficients to find , but it is not needed to answer the question.)
Compute b − c and identify the answer
Use the values of and :
So , which corresponds to answer choice C.