Question 117·Hard·Advanced Math

0 / 700

The cubic polynomial $2x^3 + ax^2 + bx + 17$ , where $a$ and $b$ are constants, can be rewritten in the form $(mx + n)(x^2 + px + q)$ , where $m$ , $n$ , $p$ , and $q$ are integers.

Which of the following expressions must be an integer?

A polynomial rewritten as a product gives a polynomial identity: the two forms agree for every $x$ . If you're asked which quotient must be an integer, isolate the constant term by setting $x=0$ . It comes from multiplying the constants of the factors, so either one divides the polynomial's constant evenly. Don't confuse a factor's coefficient of $x$ with its constant.

Hints

Hint 1
In a polynomial identity, both forms give the same result at every input. Put $x=0$ to remove all terms containing $x$ . What remains from each form?
Hint 2
The constant term is what remains when $x=0$ . On the product side, multiply the two numbers that remain. Dividing their product by one of them gives the other integer.

Step-by-step

Match the constant terms
Step 1
Set $x=0$ to isolate the constant terms
No Desmos needed. Matching constant terms, the parts without $x$ , settles this directly. The two forms agree for every $x$ , so set $x=0$ to erase every term containing $x$ :
$17=nq.$
On the product side, the surviving factors are $n$ and $q$ , not $m$ and $q$ .
Step 2
Identify the quotient that equals an integer
Since $nq=17$ and $17$ isn't zero, $n$ isn't zero. Divide by $n$ :
$\frac{17}{n}=q.$
The problem says $q$ is an integer. Matching constant terms reveals which quotient equals an integer. So $\frac{17}{n}$ must be an integer. Choice D.

Lessons that teach this

SAT Advanced AlgebraAdvancedCoreUse polynomial identities, factors, and unknown coefficients

Free Question Bank

Math

Reading & Writing

Practice sets

Take a test

One section

Before and after

Math

Reading & Writing

Desmos

The test

Your score

Read more

Study together

Challenges and chat

The guarantee

What students say

Success stories

Question 117·Hard·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this