Polynomial Expressions — SAT Math Guide

Q: What You Need to Know

• **Polynomial**: An expression with variables, coefficients, and non-negative integer exponents • **Degree**: The highest power of the variable (3x² + 5x - 2 has degree 2) • **Like terms**: Terms with the same variable and exponent that can be combined • **Standard form**: Terms arranged from highe

Polynomial expressions SAT questions test your ability to manipulate, factor, and evaluate algebraic expressions with multiple terms. These expressions involve variables raised to whole number powers, like 3x² + 5x - 2 or x³ - 4x² + 7. You'll encounter polynomial expressions in about 3-5 questions in the Advanced Math domain of the Digital SAT. With the right approach, these problems become much more manageable than they first appear.

What You Need to Know

Polynomial: An expression with variables, coefficients, and non-negative integer exponents

Degree: The highest power of the variable (3x² + 5x - 2 has degree 2)

Like terms: Terms with the same variable and exponent that can be combined

Standard form: Terms arranged from highest to lowest degree

Factoring: Rewriting a polynomial as a product of simpler expressions

FOIL method: For multiplying two binomials (First, Outer, Inner, Last)

Difference of squares: a² - b² = (a + b)(a - b)

Perfect square trinomials: a² ± 2ab + b² = (a ± b)²

📐 KEY FORMULA: (a + b)² = a² + 2ab + b²

💡 PRO TIP: Always look for common factors first before trying more complex factoring methods on the SAT.

How to Solve Polynomial Expressions on the SAT

Example Question 1 — Medium Difficulty

Which of the following is equivalent to (2x + 3)(x - 4)?

A) 2x² - 5x - 12

B) 2x² + 5x - 12

C) 2x² - 5x + 12

D) 2x² + 11x - 12

Solution:

Step 1: Use FOIL to multiply the binomials

Step 2: First: (2x)(x) = 2x², Outer: (2x)(-4) = -8x, Inner: (3)(x) = 3x, Last: (3)(-4) = -12

Step 3: Combine like terms: 2x² - 8x + 3x - 12 = 2x² - 5x - 12

✅Answer: A — FOIL gives us 2x² - 8x + 3x - 12, which simplifies to 2x² - 5x - 12.

Example Question 2 — Hard Difficulty

If x² + 6x + k = (x + p)² for all values of x, what is the value of k?

A) 3

B) 6

C) 9

D) 12

Solution:

Step 1: Expand the right side using the perfect square formula

Step 2: (x + p)² = x² + 2px + p²

Step 3: Compare coefficients: x² + 6x + k = x² + 2px + p²

Step 4: From the x terms: 6x = 2px, so p = 3

Step 5: Therefore k = p² = 3² = 9

✅Answer: C — When we expand (x + p)² and match coefficients, p = 3 and k = p² = 9.

Common SAT Math Mistakes to Avoid

❌Mistake: Forgetting to distribute the negative sign when subtracting polynomials

✅Fix: Rewrite subtraction as addition of the opposite: (3x² + 2x) - (x² - 5x) = (3x² + 2x) + (-x² + 5x)

❌Mistake: Combining unlike terms (adding x² and x terms together)

✅Fix: Only combine terms with identical variable parts and exponents

❌Mistake: Making sign errors when using FOIL, especially with negative terms

✅Fix: Write out each step of FOIL separately and double-check your signs

❌Mistake: Stopping at the first factorization without checking if you can factor further

✅Fix: Always check if your factors can be factored again, especially when you factor out a common term first

Practice Question — Try It Yourself

What is the coefficient of x² when (3x - 2)(2x² + 5x - 1) is written in standard form?

A) 6

B) 11

C) 13

D) 19

Show Answer

Answer: B — Multiply (3x)(5x) = 15x² and (-2)(2x²) = -4x², then combine: 15x² - 4x² = 11x².

Key Takeaways for the SAT

Master FOIL for multiplying binomials — it's essential for SAT math polynomial expressions

Always look for common factors before attempting complex factoring techniques

Recognize perfect square trinomials and difference of squares patterns to save time

When expanding polynomials, organize your work to avoid sign errors

Check your final answer by substituting a simple value like x = 1 when possible