ISEE Upper Level

Polynomials & Quadratics

Q: Why do some of these problems have two answers?

Because of the little `2` in `x^2`! That exponent tells you the highest number of answers possible. Since a negative times a negative makes a positive, both positive and negative numbers can solve the puzzle. 🧩

Polynomial multiplication/expansion, factoring, and solving equations of degree 2 or higher — excludes linear expression simplification (see linear-expressions-equations)

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Imagine you're building a massive Lego castle. You start with small blocks, snap them together, and suddenly you have a giant fortress! That's exactly how polynomials work in math. 🏰

'Poly' means 'many,' and 'nomial' means 'names' or 'terms.' So, a polynomial is just a math expression with many different parts stuck together, like $x^{2} + 3 x + 5$ . Quadratics are a special, super-important kind of polynomial where the highest power (the little number at the top) is a 2, like $x^{2}$ . Think of that little 2 as a VIP pass—it tells you there are usually two different answers hiding inside the puzzle!

On the ISEE, you'll get to act like a math detective. Sometimes you'll squish these blocks together (which we call 'expanding'), and sometimes you'll break them apart to see what's inside (which we call 'factoring'). Don't worry if it looks like an alien language at first. Once you learn the secret handshakes—like the famous FOIL method—you'll be solving these puzzles faster than a cheetah on roller skates! 🐆🛼

Also, here is a secret weapon for the ISEE: there is NO penalty for guessing! If a quadratic equation looks too tricky, try plugging the answer choices back into the $x$ spots to see which one works. Never leave a bubble blank, and always answer every single question!

Key Formula

The FOIL Method for multiplying binomials:

(a + b) (c + d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d

(First, Outer, Inner, Last!)

Practice Questions

5 practice questions for ISEE Upper Level

Q1 Hard

x^{2} + y^{2} = 58

and

x - y = 4

, what is the value of

x y

A 10

B 21

C 37

D 42

Show Solution

First, square both sides of the equation $x - y = 4$ to get $(x - y)^{2} = 16$ . Expanding the left side gives $x^{2} - 2 x y + y^{2} = 16$ . We are given that $x^{2} + y^{2} = 58$ . Rearranging our expanded equation, we have $(x^{2} + y^{2}) - 2 x y = 16$ . Substituting 58 into the equation yields $58 - 2 x y = 16$ . Subtracting 58 from both sides gives $- 2 x y = - 42$ . Dividing by -2 gives $x y = 21$ .

Answer: B

Q2 Hard

Which expression is equivalent to

(3 a^{2} b - 5 a b^{2}) - (a^{2} b + 2 a b^{2} - 4 ab)

2 a^{2} b - 3 a b^{2} - 4 ab

2 a^{2} b - 7 a b^{2} + 4 ab

4 a^{2} b - 3 a b^{2} + 4 ab

4 a^{2} b - 7 a b^{2} - 4 ab

Show Solution

Distribute the negative sign to each term in the second polynomial: $3 a^{2} b - 5 a b^{2} - a^{2} b - 2 a b^{2} + 4 ab$ . Next, combine the like terms. The $a^{2} b$ terms combine to $3 a^{2} b - a^{2} b = 2 a^{2} b$ . The $a b^{2}$ terms combine to $- 5 a b^{2} - 2 a b^{2} = - 7 a b^{2}$ . The $4 ab$ term does not have any like terms. The fully simplified expression is $2 a^{2} b - 7 a b^{2} + 4 ab$ .

Answer: B

Q3 Hard

What is the minimum value of

y

y = x^{2} - 4 x + 7

for

0 \leq x \leq 5

A 3

B 4

C 7

D 12

Show Solution

The graph of the quadratic equation $y = x^{2} - 4 x + 7$ is a parabola opening upwards (since the $x^{2}$ coefficient is positive), meaning its minimum value occurs at the vertex. The x-coordinate of the vertex is given by $x = \frac{- b}{2 a} = \frac{- ( - 4 )}{2 ( 1 )} = 2$ . Since $x = 2$ is within the given interval $0 \leq x \leq 5$ , we can substitute $x = 2$ into the equation to find the minimum value of $y$ : $y = (2)^{2} - 4 (2) + 7 = 4 - 8 + 7 = 3$ .

Answer: A

Q4 Hard

Compare the quantities in Column A and Column B.

Column A

(p + q)^{2} - (p - q)^{2}

Column B

4 pq

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

Expand the binomial expressions in Column A. $(p + q)^{2} = p^{2} + 2 pq + q^{2}$ and $(p - q)^{2} = p^{2} - 2 pq + q^{2}$ . Subtracting the second from the first gives $(p^{2} + 2 pq + q^{2}) - (p^{2} - 2 pq + q^{2})$ . Distribute the negative sign and combine like terms: $p^{2} + 2 pq + q^{2} - p^{2} + 2 pq - q^{2} = 4 pq$ . Since Column A simplifies exactly to $4 pq$ , which is identical to Column B, the two quantities are always equal regardless of the values of $p$ and $q$ .

Answer: C

Q5 Hard

x^{2} - k x + 36

is a perfect square trinomial and

k

is a positive integer, what is the value of

k

A 6

B 12

C 18

D 36

Show Solution

A perfect square trinomial takes the form $(x - a)^{2} = x^{2} - 2 a x + a^{2}$ or $(x + a)^{2} = x^{2} + 2 a x + a^{2}$ . In the given expression $x^{2} - k x + 36$ , the constant term is $a^{2} = 36$ , which means $a = 6$ or $a = - 6$ . The middle term is $- k x$ , which must correspond to either $- 2 (6) x$ or $2 (6) x$ . Therefore, $- k = - 12$ or $- k = 12$ , meaning $k = 12$ or $k = - 12$ . Since the problem states $k$ is a positive integer, $k$ must be 12.

Answer: B

Tips & Strategies

On ISEE Quantitative Comparison questions, don't try to find the exact values of $x$ and $y$ if you don't have to! Look for ways to use the whole chunk of information, like substituting $2 x y$ all at once.
Always check your middle terms when using FOIL. A super common trick on the ISEE is giving answer choices where the middle terms are added instead of subtracted. Take your time combining them!
If you solve a quadratic and get a fraction like $x = \frac{1}{2}$ , don't panic! You can plug fractions back into the original equation just like whole numbers to check your work.

Common Mistakes

Watch out for the 'distributing the square' trap! $(x + y)^{2}$ is NOT $x^{2} + y^{2}$ . You have to write it out as $(x + y) (x + y)$ and use FOIL. It's like trying to bake a cake without mixing the ingredients first! 🎂
Don't forget the evil twin! When you see $x^{2} = 25$ , remember that every positive answer usually has a negative twin. The answer is $5$ AND $- 5$ . Don't leave the negative twin behind!

Frequently Asked Questions

What does FOIL stand for?

FOIL is a handy memory trick for multiplying two binomials. It stands for First, Outer, Inner, Last. It makes sure you multiply every part of the first group by every part of the second group!

Why do some of these problems have two answers?

Because of the little $2$ in $x^{2}$ ! That exponent tells you the highest number of answers possible. Since a negative times a negative makes a positive, both positive and negative numbers can solve the puzzle. 🧩

Should I guess if I get stuck on a hard polynomial question?

Yes, absolutely! The ISEE does NOT penalize you for wrong answers. If you're stuck, try to eliminate one or two silly choices, pick your favorite letter, and move on. 🚀

What is a Quantitative Comparison question?

It's a special puzzle on the ISEE where you compare Column A and Column B. You don't always have to solve everything perfectly. You just need to figure out which side is bigger, or if they are equal!

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