AMC 10 · 2017 · #23
Grade 8 algebraPick an answer.
f shares all three roots of g and has exactly one more root. Since both are monic, f must equal g times one linear factor (x - r). Name that fourth root r, expand the product, then match it term-by-term against the given f. Matching coefficients turns the problem into small linear equations that pin down r, a, b, and c.
Write f as g times a linear factor
The quartic is the cubic times a linear factor.
A monic quartic that contains a monic cubic's roots is just that cubic times one leftover (x - r).
6.EE.B.6Introduce A VariableExpand the product
Expanding writes every coefficient explicitly.
Distributing turns the factored form into one whose coefficients we can compare directly.
6.EE.A.3Introduce A VariableMatch coefficients to find r
One coefficient names the extra root.
Two equal polynomials must agree coefficient by coefficient, so each power gives its own little equation.
Two equal polynomials must agree coefficient by coefficient, so each power gives its own little equation.
▸ Why?
When two expressions are equal for every input, each matching coefficient must be equal too.
▸ Why?
The factored form is legitimate because a polynomial vanishing at a root carries that root's linear factor.
Find a, c, then b
The rest then follow immediately.
Once r is known, every other coefficient drops out by plugging back into its own equation.
7.NS.A.2Identify SubproblemsEvaluate f(1)
Evaluating gives -7007, choice (B).
f(1) is just the sum of f's coefficients, and the factored form gives a clean second route to the same number.
6.EE.A.2Introduce A VariableWhen one monic polynomial holds all the roots of a smaller monic one, it is just the smaller one times a single (x - r) leftover, and matching coefficients reveals everything.
- Write f as g times a linear factor
- Expand the product
- Match coefficients to find r
- Find a, c, then b
- Evaluate f(1)