AMC 10 · 2009 · #9
Grade 9 algebraPick an answer.
Chasing a, b, c one at a time is not required. The sum a+b+c is exactly what the standard form returns at x = 1, so change focus from three unknown coefficients to the single number f(1). The given rule reports f at inputs of the form x+3, so work backwards to the x that makes x+3 equal 1. Then rebuild a, b, c by algebra as an independent route that also confirms such an f exists.
Read a+b+c as one output
The coefficient sum is the value at one.
At x = 1 every power of the variable is 1, so a polynomial's value is just the sum of its coefficients.
At an input of one every power collapses to one, so the value is just the sum of the coefficients.
▸ Why?
One raised to any power is still one, so each coefficient stands alone with nothing multiplying it.
▸ Why?
The polynomial's value is its terms added together, so with the powers gone only the coefficients remain.
Work backwards to the right input
Working backwards through the shift finds the right input.
The formula is a machine labelled by x+3; to read its answer at 1, feed it x = -2.
8.EE.C.7Work BackwardsEvaluate the given quadratic at x = -2
Evaluating there gives 2.
One well-chosen input turns the whole identity into a single line of arithmetic.
9.F-IF.A.2Work BackwardsConfirm such an f really exists
Such a quadratic really exists.
Agreeing at every x is far stronger than agreeing at one x — it pins the coefficients down one by one.
9.A-SSE.A.2Convert To AlgebraSolve for a, b, c and compare
Solving for the coefficients confirms 2, choice (A).
Building f explicitly shows the shortcut was no coincidence: the quadratic is unique, so its coefficient sum is too.
8.EE.C.7Convert To AlgebraThe sum of a polynomial's coefficients is just its value at x = 1, so find the one input that makes the given rule report f(1).
- Read a+b+c as one output
- Work backwards to the right input
- Evaluate the given quadratic at x = -2
- Confirm such an f really exists
- Solve for a, b, c and compare