Question 28·200 Super-Hard SAT Math Questions·Advanced Math

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f(x)=x+p

g(x)=3x^2-qx+r

The functions $f$ and $g$ are given, where $p$ is a positive integer and $q$ and $r$ are integers. If

f(x)\cdot g(x)=3x^3+4x^2-73x-90

for all $x$ , which choice is the value of $p$ ?

A polynomial identity holds at every input when the problem says for all $x$ . A factor $x+p$ has a root at $-p$ . Graph the cubic in Desmos to find a negative integer root, then evaluate it to check that it is exactly zero. Don’t mistake $p$ for the root; $x+p$ equals zero when $x=-p$ .

Hints

Hint 1
An identity is an equation true for every input. Find the input that makes $f(x)=x+p$ equal $0$ . What must the cubic equal at that same input?
Hint 2
An $x$ -intercept is where a graph meets the horizontal axis, so the polynomial equals $0$ there. Since $p$ is a positive integer, look for an intercept at a negative integer.
Hint 3
A graph can suggest an integer root without proving it is exact. Evaluate the cubic at that input in Desmos. If the result is $0$ , use the root $-p$ to find $p$ .

Step-by-step

Find the root of the given factor
Step 1
Connect the factor to a root
Set the given factor $x+p$ equal to zero:
$x+p=0.$
Subtract $p$ :
$x=-p.$
Because the product equals the cubic for all $x$ , the cubic must also be zero at $-p$ . A root is an input that makes a polynomial zero, so you need a negative integer root.
Step 2
Find the negative integer root
Type $P(x)=3x^3+4x^2-73x-90$ in Desmos. Click the negative $x$ -intercepts: Desmos shows $(-5,0)$ and approximately $(-1.23,0)$ . The second input isn't an integer, so it can't equal $-p$ . The intercept at $-5$ is the candidate to check exactly.
Step 3
Confirm the root and find $p$
Add $P(-5)$ in Desmos. It prints $0$ , confirming that $-5$ is an exact root. Set it equal to the factor's root:
$-p=-5.$
Multiply by $-1$ :
$p=5.$
This is a positive integer, as required. Choice C.

Question 28·200 Super-Hard SAT Math Questions·Advanced Math

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