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SAT Digital Math · SL AA

Advanced Math: Equivalent Expressions and Polynomials

Targeted Practice Sheet 2

Worked Examples: SAT Strategy

Example 1: Easy (Matching Coefficients)

Question: If $3x^2 + kx - 8 = (3x + 4)(x - 2)$ for all values of $x$, what is the value of $k$?
SAT Strategy: The phrase "for all values of $x$" means the left side and right side are mathematically identical. Just expand the right side and match the coefficients!
  1. Expand the right side using FOIL: $3x(x) + 3x(-2) + 4(x) + 4(-2)$.
  2. Simplify: $3x^2 - 6x + 4x - 8$.
  3. Combine like terms: $3x^2 - 2x - 8$.
  4. Match the $x$-coefficient to the left side: $kx = -2x$, so $k = -2$.
  5. Answer: $-2$.

Example 2: Medium (Rational Exponents)

Question: Which of the following expressions is equivalent to $\frac{\sqrt{x^5}}{\sqrt{x}}$ for $x > 0$?
SAT Strategy: Convert all radicals to rational exponents ($x^{a/b}$), then apply standard exponent subtraction rules ($\frac{x^c}{x^d} = x^{c-d}$).
  1. Rewrite the numerator: $\sqrt{x^5} = x^{5/3}$.
  2. Rewrite the denominator: $\sqrt{x} = x^{1/2}$.
  3. Set up the division: $\frac{x^{5/3}}{x^{1/2}}$.
  4. Subtract the exponents: $\frac{5}{3} - \frac{1}{2}$. Find a common denominator (6).
  5. $\frac{10}{6} - \frac{3}{6} = \frac{7}{6}$.
  6. Answer: $x^{7/6}$ (or $\sqrt{x^7}$).

Example 3: Hard (The Remainder Theorem Shortcut)

Question: The expression $\frac{5x + 14}{x + 2}$ can be rewritten in the form $A + \frac{B}{x + 2}$, where $A$ and $B$ are constants. What is the value of $B$?
SAT Strategy: Most students use polynomial long division here. As an AA SL student, you know the Remainder Theorem! The constant $B$ over the divisor is just the remainder. To find the remainder when dividing by $(x - a)$, simply evaluate the polynomial at $x = a$.
  1. The divisor is $(x + 2)$, which means our root to test is $x = -2$.
  2. The numerator is our polynomial $P(x) = 5x + 14$.
  3. Evaluate $P(-2)$ to find the remainder $B$: $5(-2) + 14$.
  4. $-10 + 14 = 4$.
  5. Therefore, the remainder $B = 4$. (And $A$ is obviously 5, the quotient).
  6. Answer: $4$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. Which expression is equivalent to $(2x^2 - 4x + 3) - (x^2 - 5x - 2)$?
    • A) $x^2 - 9x + 1$
    • B) $x^2 + x + 5$
    • C) $3x^2 - 9x + 1$
    • D) $x^2 - x + 5$
  2. Which expression is equivalent to $4x^3 \cdot 2x^5$?
    • A) $6x^8$
    • B) $8x^{15}$
    • C) $8x^8$
    • D) $6x^{15}$
  3. If $(ax + 2)(bx - 3) = 15x^2 + cx - 6$ for all values of $x$, and $a + b = 8$, what are the two possible values for $c$?
    • A) $1$ and $-1$
    • B) $-1$ and $-19$
    • C) $3$ and $5$
    • D) $1$ and $-9$
  4. Which expression is equivalent to $(x^2y^3)^4$?
    • A) $x^6y^7$
    • B) $x^8y^{12}$
    • C) $x^{16}y^{81}$
    • D) $x^2y^{12}$
  5. What is the value of $(3x - 2)^2$ when expanded?
    • A) $9x^2 - 4$
    • B) $9x^2 + 4$
    • C) $9x^2 - 12x + 4$
    • D) $9x^2 - 6x + 4$

Level 2: Medium (Standard SAT)

  1. Which expression is equivalent to $\frac{x^2 - 16}{x^2 + 8x + 16}$ for $x \neq -4$?
    • A) $-1$
    • B) $\frac{x - 4}{x + 4}$
    • C) $\frac{x + 4}{x - 4}$
    • D) $\frac{1}{8x}$
  2. If $3^x = 10$, what is the value of $3^{x+2}$? (Grid-in)
  3. Which of the following is equivalent to $\sqrt{x^3}$?
    • A) $x^{3/4}$
    • B) $x^{4/3}$
    • C) $x^7$
    • D) $x^{-3/4}$
  4. If $\frac{x^{a^2}}{x^{b^2}} = x^{24}$, $x > 1$, and $a + b = 4$, what is the value of $a - b$? (Grid-in)
  5. The expression $\frac{6x + 11}{2x + 1}$ can be rewritten as $3 + \frac{k}{2x + 1}$, where $k$ is a constant. What is the value of $k$? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. Which of the following is equivalent to $\frac{1}{\frac{1}{x} + \frac{1}{y}}$?
    • A) $x + y$
    • B) $\frac{x + y}{xy}$
    • C) $\frac{xy}{x + y}$
    • D) $xy$
  2. A polynomial $P(x)$ is divided by $(x - 3)$. The quotient is $x^2 - 4x + 2$ and the remainder is $7$. What is $P(3)$? (Grid-in)
  3. Which of the following is a factor of $x^3 - 2x^2 - 9x + 18$?
    • A) $x - 1$
    • B) $x + 2$
    • C) $x + 3$
    • D) $x - 4$
  4. If $8^x \cdot 2^y = 64$ and $3x + y = k$, what is the value of $k$? (Grid-in)
  5. The expression $(x^2 - y^2)^2 - (x^2 + y^2)^2$ is equivalent to which of the following?
    • A) $0$
    • B) $-4x^2y^2$
    • C) $4x^2y^2$
    • D) $-2y^4$

Level 4: Very Hard (Slightly beyond SAT)

  1. The equation $\frac{24x^2 + 25x - 47}{ax - 2} = -8x - 3 - \frac{53}{ax - 2}$ is true for all values of $x \neq \frac{2}{a}$, where $a$ is a constant. What is the value of $a$? (Grid-in)
  2. If $x > 0$ and $x^{1/2} \cdot x^{1/3} \cdot x^{1/6} = x^n$, what is the value of $n$? (Grid-in)
  3. A rational expression simplifies as follows: $\frac{cx^2 + 10x - 8}{2x + 4} = 3x - 2$ for $x \neq -2$. What is the value of $c$? (Grid-in)
  4. Let $p(x) = x^3 - ax^2 + bx - 6$. If $(x - 1)$ and $(x + 2)$ are both factors of $p(x)$, what is the value of $a + b$? (Grid-in)
  5. The expression $\frac{1}{x-1} + \frac{2}{x+2}$ is equivalent to $\frac{cx + d}{x^2 + x - 2}$. What is the value of $c \times d$? (Grid-in)

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