Quadratics and Polynomials 1 Bank

Quadratics and Polynomials — SAT Tips, Tricks, and Why They Matter

Why this matters on the SAT

Quadratics and polynomials show up often in SAT Math, especially in Advanced Math. These questions test whether you can:

• factor
• solve equations
• find zeros / roots
• rewrite expressions
• use structure instead of brute force
• connect equations, graphs, and solutions

The SAT loves to test whether you recognize that a problem is really about structure.
A messy-looking expression often becomes much easier once you spot a pattern.

Core things to remember

1. A quadratic usually looks like

ax² + bx + c

Examples:

• x² + 5x + 6
• 2x² – 7x + 3
• x² – 16

A polynomial can be bigger, but on the SAT, quadratics are the star.

2. The most important SAT skill: factoring

If you can factor quickly, you unlock a lot of problems.

Common patterns

Standard trinomial

• x² + 5x + 6 = (x + 2)(x + 3)

Difference of squares

• x² – 16 = (x – 4)(x + 4)

Take out the greatest common factor first

• 3x² + 12x = 3x(x + 4)

SAT trick

Always check for a GCF first before doing anything else.
A lot of students miss easy factoring because they jump too fast.

3. Zero product property

If:
(x – 5)(x + 2) = 0

then:

• x – 5 = 0 or
• x + 2 = 0

So the solutions are:
x = 5 and x = -2

Important

A product equals zero only if at least one factor equals zero.
That idea shows up constantly.

4. Solutions, roots, zeros, x-intercepts

On the SAT, these often mean the same basic thing.

If the question asks for:

• solutions
• roots
• zeros
• x-values where the graph crosses the x-axis

you are usually looking for the values of x that make the expression equal 0.

Example:
x² – 9 = 0
Factor: (x – 3)(x + 3) = 0
So the zeros are 3 and -3

Big SAT idea

Set the equation equal to 0 before solving, if it is not already.

5. Not every problem wants both answers

Sometimes the SAT asks for:

• the greater solution
• the positive solution
• the least value
• the solution greater than 0

So even if you find two answers, read carefully before choosing.

Example:
If the solutions are 5 and -2, and the question asks for the greater solution, the answer is 5.

6. Quadratics can be solved in more than one way

You may use:

• factoring
• square roots
• completing the square
• quadratic formula

On the SAT, the best move is usually the fastest clean method.

Use square roots when

The equation looks like:
(x – 3)² = 49

Then:
x – 3 = 7 or x – 3 = -7
So:
x = 10 or x = -4

Very important

When taking square roots, remember the plus and minus.

√49 = 7, but solving x² = 49 gives
x = 7 and x = -7

That mistake costs points all the time.

7. Difference of squares is a favorite

If you see:
x² – a²

think:
(x – a)(x + a)

Examples:

• x² – 25 = (x – 5)(x + 5)
• x² – 81 = (x – 9)(x + 9)

SAT shortcut

If you recognize this immediately, you save a lot of time.

8. Watch for quadratics disguised as something simpler

Example:
12x² – 42x = 0

Many students panic because it looks ugly.
But first factor out the GCF:

6x(2x – 7) = 0

Then:

• 6x = 0 → x = 0
• 2x – 7 = 0 → x = 7/2

Important

Do not skip the factoring-out step.
That is often the whole question.

9. The SAT loves equivalent expressions

Sometimes the question is not “solve.”
Sometimes it asks which expression is equivalent or which is a factor.

Example:
Which is a factor of x² + 2x – 24?

Factor:
(x + 6)(x – 4)

So a factor is x + 6

SAT trick

If the answers are factor-looking expressions, factor the polynomial fully before guessing.

10. Graph connections matter

A quadratic graph is called a parabola.

Things the SAT may connect:

• zeros / x-intercepts
• vertex
• maximum or minimum
• factored form vs standard form

Quick connection

If a quadratic is written as:
(x – 2)(x + 5)

the zeros are:

• x = 2
• x = -5

That means the graph crosses the x-axis at 2 and -5.