Question 718·Hard·Math

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The function

f(x)=x^3+ax^2+bx+c

is defined above, where $a$ , $b$ , and $c$ are real constants. The function has a zero at $x=2$ of multiplicity 2. If $f(1)=6$ , what is the value of $c$ ?

Your answer

A zero of multiplicity 2 gives two copies of the same factor. For a cubic with leading coefficient $1$ , one linear factor remains unknown; use the known output to find it with a Desmos regression. Then evaluate at $0$ to get the constant term. An output at $1$ is not the constant term.

Hints

Hint 1
Multiplicity 2 means a zero supplies two copies of its factor. A zero at $2$ supplies $(x-2)^2$ ; what linear factor must be left in a cubic?
Hint 2
Write the cubic as $(x-2)^2(x+d)$ . To use $f(1)=6$ , replace $x$ with $1$ and set the product equal to $6$ . This gives one equation for $d$ .
Hint 3
The constant term survives when $x=0$ , because every term containing $x$ disappears. After finding $d$ , evaluate the factored form at $0$ , not at $1$ .

Step-by-step

Build the cubic from its repeated zero
Step 1
Write the repeated factor
A zero is an input that makes the function equal $0$ . Multiplicity 2 means the factor for the zero at $2$ appears twice, giving $(x-2)^2$ . That accounts for two of the cubic’s three factors. Its $x^3$ coefficient is $1$ , so write the remaining factor as $x+d$ without a number out front: $f(x)=(x-2)^2(x+d)$ . Checking only $f(2)=0$ would establish one factor of $x-2$ , not two.
Step 2
Turn the known output into an equation
The statement $f(1)=6$ means put $1$ into the factored rule and make the result $6$ : $(1-2)^2(1+d)=6$ . This equation uses the output at $1$ to find $d$ ; it does not say that $c=6$ .
Step 3
Find the remaining factor in Desmos
Type $(1-2)^2(1+d)\sim6$ . The symbol $\sim$ tells Desmos to fit the unknown constant $d$ . Under PARAMETERS, it shows $d=5$ , so the remaining factor is $x+5$ .
Step 4
Evaluate at zero to get the constant term
The constant term $c$ is $f(0)$ : at $x=0$ , the $x^3$ , $ax^2$ , and $bx$ terms all disappear. Keep the regression, then type $f(x)=(x-2)^2(x+d)$ and $f(0)$ . Desmos uses $d=5$ and prints $20$ , so $c=20$ . Grid in 20.

Question 718·Hard·Math

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