SSAT Upper Level

Polynomials & Quadratics

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 trying to build the ultimate ice cream sundae. You have scoops of ice cream, squirts of chocolate syrup, and colorful sprinkles. Polynomials are just like that! "Poly" means many, and "nomial" means terms. So a polynomial is just a math sundae with many parts (terms) added together, like $x^{2}$ , $3 x$ , and $5$ . When the biggest exponent in your math sundae is a 2 (like $x^{2}$ ), we call it a "quadratic." Think of the $x^{2}$ as the giant cherry on top! 🍒

On the SSAT, you'll be asked to play around with these math sundaes. Sometimes you'll "expand" them, which is like mixing all the ingredients together. Other times you'll "factor" them, which is like un-mixing the sundae back into separate scoops and toppings (magic, right?). Don't let the fancy words scare you. If you know how to multiply and follow the rules of a puzzle, you are going to totally rock polynomials and quadratics! 🚀

When you "solve" a quadratic equation, you're basically playing detective. 🕵️‍♂️ You're looking for the secret identity of $x$ that makes the whole equation equal zero. It's like finding the exact number of sprinkles needed to balance a giant scale perfectly. The SSAT loves to test your pattern-spotting skills! So keep your eyes peeled for special equations that look like twins or opposites. Once you learn the secret handshakes of quadratics, these questions become some of the fastest and most fun puzzles on the whole test!

Key Formula

The Perfect Square Formula:

(x + y)^{2} = x^{2} + 2 x y + y^{2}

Practice Questions

5 practice questions for SSAT Upper Level

Q1 Hard

What is the greatest common factor of

15 a^{3} b^{2}

and

25 a^{2} b^{4}

5 ab

5 a^{2} b^{2}

15 a^{2} b^{2}

75 a^{3} b^{4}

75 a^{5} b^{6}

Show Solution

First, find the greatest common factor of the coefficients, 15 and 25, which is 5. Next, find the greatest common factor of the variables by taking the lowest exponent for each variable present in both terms. For $a^{3}$ and $a^{2}$ , the greatest common factor is $a^{2}$ . For $b^{2}$ and $b^{4}$ , the greatest common factor is $b^{2}$ . Combining these parts gives the overall greatest common factor: $5 a^{2} b^{2}$ .

Answer: B

Q2 Hard

Which of the following is a factor of

x^{2} - 7 x + 12

x + 3

x - 3

x + 4

x - 7

x + 12

Show Solution

To factor the quadratic expression $x^{2} - 7 x + 12$ , you must find two numbers that multiply to the constant term (12) and add to the coefficient of the middle term (-7). These numbers are -3 and -4, because (-3)(-4) = 12 and -3 + (-4) = -7. Therefore, the quadratic factors into $(x - 3) (x - 4)$ . Of the given choices, only $x - 3$ is a factor.

Answer: B

Q3 Hard

2 (x + 4) (x - 5) = 0

Which of the following values of

x

satisfies the equation above?

A 5

B 4

C 2

D -2

E -5

Show Solution

For the product of the factors to equal zero, at least one of the factors must be equal to zero. Setting each variable factor to zero gives $x + 4 = 0$ (which means $x = - 4$ ) and $x - 5 = 0$ (which means $x = 5$ ). The value 5 is among the answer choices, satisfying the equation.

Answer: A

Q4 Hard

Which of the following is equivalent to

(2 x + 3) (x^{2} - x)

2 x^{3} - 3 x

2 x^{3} - x^{2} - 3 x

2 x^{3} + x^{2} - 3 x

2 x^{3} + 5 x^{2} - 3 x

2 x^{3} - 3 x^{2} - 3 x

Show Solution

To find the equivalent expression, distribute each term in the first binomial to each term in the second binomial. Multiply $2 x$ by $x^{2}$ to get $2 x^{3}$ . Multiply $2 x$ by $- x$ to get $- 2 x^{2}$ . Multiply $3$ by $x^{2}$ to get $3 x^{2}$ . Multiply $3$ by $- x$ to get $- 3 x$ . This gives $2 x^{3} - 2 x^{2} + 3 x^{2} - 3 x$ . Finally, combine the like terms $- 2 x^{2}$ and $3 x^{2}$ to get $x^{2}$ . The resulting simplified expression is $2 x^{3} + x^{2} - 3 x$ .

Answer: C

Q5 Hard

Which of the following is equivalent to

9 x^{2} - 24 x y + 16 y^{2}

(3 x + 4 y)^{2}

(3 x - 4 y)^{2}

(9 x - 16 y)^{2}

(3 x - 4 y) (3 x + 4 y)

3 (x - 4 y)^{2}

Show Solution

The given expression is a perfect square trinomial. The first term, $9 x^{2}$ , is the square of $3 x$ . The third term, $16 y^{2}$ , is the square of $4 y$ . The middle term, $- 24 x y$ , is equal to $2 (3 x) (- 4 y)$ . This matches the algebraic pattern $a^{2} - 2 ab + b^{2} = (a - b)^{2}$ . Therefore, the expression is equivalent to $(3 x - 4 y)^{2}$ .

Answer: B

Tips & Strategies

💡 Always look to plug in numbers first! If a question asks for the value of a polynomial and gives you $x$ , don't multiply the letters together. Just plug the number in immediately to save time.
⏱️ Memorize the 'Difference of Squares' pattern: $x^{2} - y^{2} = (x + y) (x - y)$ . The SSAT test-makers love hiding this pattern in upper-level math questions!
🎯 When factoring a quadratic equation to solve for $x$ , always make sure the equation is set to equal 0 first (like $x^{2} + 3 x - 4 = 0$ ). If it doesn't equal zero, you can't use your factoring tricks!

Common Mistakes

👻 Watch out for the 'phantom negative' mistake! When you see $x^{2} = 16$ , many students only answer $x = 4$ . Don't forget that $x = - 4$ is also a solution because $- 4 \cdot - 4 = 16$ .
🪤 Don't fall for the $(x + y)^{2}$ trap! A lot of students think $(x + y)^{2}$ is just $x^{2} + y^{2}$ . Nope! You have to multiply the whole thing out, which gives you a middle term too: $x^{2} + 2 x y + y^{2}$ .

Frequently Asked Questions

What does 'quadratic' even mean?

🏛️ 'Quadratic' just means the biggest exponent in the equation is a 2 (like $x^{2}$ ). The word comes from 'quadratum,' which is Latin for square!

Do I need to know the Quadratic Formula for the SSAT?

🧠 Nope! The SSAT almost never requires the big, complicated Quadratic Formula ( $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ ). Usually, you can solve the equations by simple factoring or plugging in the answer choices!

What is FOIL?

FOIL is a fun memory trick for multiplying two binomials, like $(x + 2) (x + 3)$ . It stands for First, Outer, Inner, Last. It reminds you exactly which pieces to multiply together so you don't miss any!

How many polynomial questions are on the SSAT?

If you are taking the Upper Level SSAT, you will usually see 2 to 4 questions about polynomials and quadratics. They are great places to pick up points once you know the basic patterns!

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