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SAT Math20 cards

Advanced Math Flashcards

Advanced Math covers polynomial operations, exponential and radical expressions, rational equations, complex numbers, and nonlinear functions tested on the SAT. These topics typically appear in the harder questions and can separate a good score from a great one.

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How do you add or subtract polynomials?

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Combine like terms (same variable and exponent). Example: (3x^2 + 2x) + (x^2 - 5x) = 4x^2 - 3x.

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How do you multiply two binomials using FOIL?

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First, Outer, Inner, Last. (a + b)(c + d) = ac + ad + bc + bd. Example: (x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6.

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What is the difference of squares formula?

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a^2 - b^2 = (a + b)(a - b). Recognize this pattern to factor quickly.

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What is a perfect square trinomial?

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a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2.

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How do you simplify expressions with exponents when multiplying?

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Add the exponents: x^a * x^b = x^(a+b). Example: x^3 * x^4 = x^7.

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How do you simplify expressions with exponents when dividing?

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Subtract the exponents: x^a / x^b = x^(a-b). Example: x^5 / x^2 = x^3.

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What does a negative exponent mean?

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x^(-n) = 1/x^n. A negative exponent flips the base to the denominator.

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What does a fractional exponent mean?

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x^(m/n) = the nth root of x^m. Example: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.

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How do you solve a radical equation like sqrt(x + 5) = 3?

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Square both sides: x + 5 = 9, so x = 4. Always check by plugging back in: sqrt(4 + 5) = sqrt(9) = 3. It works.

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What is a complex number?

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A number in the form a + bi, where i = sqrt(-1). Example: 3 + 2i. The SAT tests basic operations with complex numbers.

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What is i^2?

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i^2 = -1. This is the defining property of the imaginary unit.

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How do you find the remainder when dividing polynomials?

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Use the Remainder Theorem: the remainder of f(x) / (x - a) equals f(a). Plug in the value instead of doing long division.

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What is the end behavior of a polynomial with a positive leading coefficient and even degree?

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Both ends go up (toward positive infinity). As x goes to positive or negative infinity, f(x) goes to positive infinity.

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How do you solve an exponential equation like 2^x = 16?

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Rewrite both sides with the same base: 2^x = 2^4, so x = 4. If bases don't match, use logarithms.

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What is exponential growth vs. exponential decay?

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Growth: y = a(1 + r)^t where r > 0. Decay: y = a(1 - r)^t where 0 < r < 1. The base determines direction.

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How do you solve a rational equation like 1/x + 1/3 = 1/2?

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Find the LCD (6x), multiply every term by it, then solve the resulting equation. Check that your answer does not make any denominator zero.

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What are the zeros of a polynomial?

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The x-values where the polynomial equals zero (where the graph crosses or touches the x-axis). Found by setting f(x) = 0 and solving.

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How do you solve absolute value equations like |2x - 1| = 7?

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Set up two cases: 2x - 1 = 7 (so x = 4) and 2x - 1 = -7 (so x = -3). Both are valid solutions.

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What is the vertex form of a quadratic?

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y = a(x - h)^2 + k, where (h, k) is the vertex. If a > 0, the parabola opens up; if a < 0, it opens down.

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How does completing the square work?

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For x^2 + bx, add (b/2)^2 to create a perfect square trinomial. Example: x^2 + 6x becomes x^2 + 6x + 9 = (x + 3)^2. Remember to balance both sides.

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Study Tips for Advanced Math

1

Master exponent rules first. They appear in almost every advanced math question and mistakes here cascade through the problem.

2

For polynomial questions, the Remainder Theorem and Factor Theorem save time. If f(a) = 0, then (x - a) is a factor.

3

Complex number problems on the SAT are usually straightforward arithmetic. Remember that i^2 = -1 and treat i like a variable until you can simplify.

4

Practice recognizing special factoring patterns (difference of squares, perfect square trinomials). The SAT rewards pattern recognition over brute-force calculation.

Frequently Asked Questions

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