Worked Examples: SAT Strategy
Example 1: Easy (Adding/Subtracting Polynomials)
Question: Which expression is equivalent to $(2x^2 - 4x + 1) - (x^2 - 5x - 3)$?
SAT Strategy: The most common mistake is forgetting to distribute the negative sign to
every term in the second polynomial.
- Distribute the negative: $2x^2 - 4x + 1 - x^2 + 5x + 3$.
- Group like terms together: $(2x^2 - x^2) + (-4x + 5x) + (1 + 3)$.
- Simplify: $x^2 + x + 4$.
- Answer: $x^2 + x + 4$.
Example 2: Medium (Factoring Rational Expressions)
Question: Which expression is equivalent to $\frac{x^2 - 16}{x - 4}$ for $x \neq 4$?
SAT Strategy: Do not attempt to plug in numbers or use a calculator. Always look for the "Difference of Squares" pattern ($a^2 - b^2 = (a-b)(a+b)$) or standard quadratic factoring to cancel out the denominator.
- Factor the numerator (Difference of Squares): $x^2 - 16 = (x - 4)(x + 4)$.
- Rewrite the fraction: $\frac{(x - 4)(x + 4)}{x - 4}$.
- Cancel the common factor of $(x - 4)$ from the top and bottom.
- Answer: $x + 4$.
Example 3: Hard (Polynomial Division Concept)
Question: The expression $\frac{3x + 14}{x + 4}$ can be rewritten as $3 + \frac{R}{x + 4}$. What is the value of the constant $R$?
SAT Strategy: You want to manipulate the numerator so that it contains a perfect multiple of the denominator.
- The denominator is $x + 4$. To factor out a whole number $3$, we need $3(x + 4) = 3x + 12$ in the numerator.
- Rewrite the original numerator $3x + 14$ by splitting the constant: $(3x + 12) + 2$.
- Split the fraction into two parts: $\frac{3x + 12}{x + 4} + \frac{2}{x + 4}$.
- Simplify the first fraction: $3 + \frac{2}{x + 4}$.
- Compare this to the requested format $3 + \frac{R}{x+4}$. It is clear that $R = 2$.
- Answer: $2$.
Practice Questions
Level 1: Easy (Foundational SAT)
- Which expression is equivalent to $(3x^2 - 5x + 2) + (x^2 + 4x - 6)$?
- A) $4x^2 - x - 4$
- B) $4x^2 - 9x - 4$
- C) $3x^4 - x^2 - 12$
- D) $2x^2 - x + 8$
- Which expression is equivalent to $4x(2x^2 - 3x + 1)$?
- A) $8x^3 - 12x^2 + 4x$
- B) $6x^3 - 7x^2 + 5x$
- C) $8x^2 - 12x + 4$
- D) $8x^3 - 3x + 1$
- Which expression is equivalent to $(x + 6)(x - 6)$?
- A) $x^2 - 12x - 36$
- B) $x^2 + 12x - 36$
- C) $x^2 - 36$
- D) $x^2 + 36$
- Which expression is equivalent to $(5x^3 - 2x) - (3x^3 - 2x)$?
- A) $2x^3 - 4x$
- B) $2x^3$
- C) $8x^3 - 4x$
- D) $8x^3$
- The expression $2(x+3)^2$ is equivalent to which of the following?
- A) $2x^2 + 18$
- B) $2x^2 + 9$
- C) $2x^2 + 6x + 18$
- D) $2x^2 + 12x + 18$
Level 2: Medium (Standard SAT)
- Which expression is equivalent to $\frac{12x^3 - 8x^2 + 4x}{4x}$, where $x \neq 0$?
- A) $3x^2 - 2x$
- B) $3x^2 - 2x + 1$
- C) $8x^2 - 4x + 1$
- D) $3x^3 - 2x^2 + x$
- Which expression is equivalent to $(2x - 5)(3x + 4)$?
- A) $6x^2 - 20$
- B) $6x^2 - 7x - 20$
- C) $6x^2 + 7x - 20$
- D) $6x^2 - 23x - 20$
- Which expression is equivalent to $\frac{x^2 - 25}{x - 5}$ for $x \neq 5$?
- A) $x - 5$
- B) $x + 5$
- C) $x - 20$
- D) $x^2 - 5$
- Which expression is equivalent to $x^2y - 3xy^2 + 2x^2y$?
- A) $3x^2y - 3xy^2$
- B) $-xy^2$
- C) $3x^4y^2 - 3xy^2$
- D) $0$
- The expression $3x^2 + 21x + 30$ can be rewritten as $3(x+a)(x+b)$, where $a$ and $b$ are positive constants. What is the value of $a \times b$? (Grid-in)
Level 3: Hard (Advanced SAT)
- Which expression is equivalent to $\frac{42a}{k} + 42ak$, where $k > 0$?
- A) $\frac{84a}{k}$
- B) $\frac{84ak^2}{k}$
- C) $\frac{42a(k+1)}{k}$
- D) $\frac{42a(k^2+1)}{k}$
- If $ax^2 + bx + c = (2x - 3)(5x + 1)$ for all values of $x$, what is the value of $b$? (Grid-in)
- Which expression is equivalent to $\frac{2x^2 + 7x + 5}{x+1}$ for $x \neq -1$?
- A) $2x + 5$
- B) $2x + 7$
- C) $2x - 5$
- D) $x + 5$
- The expression $\frac{x}{y} + \frac{y}{x}$ is equivalent to which of the following?
- A) $1$
- B) $\frac{x+y}{xy}$
- C) $\frac{x^2+y^2}{xy}$
- D) $\frac{(x+y)^2}{xy}$
- If $(x - 4)^2 - 16 = x^2 - px$ for all values of $x$, what is the value of $p$? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- The expression $2x^2 + bx - 45$, where $b$ is a constant, can be rewritten as $(hx+k)(x+j)$, where $h$, $k$, and $j$ are integer constants. Which of the following must be an integer?
- A) $\frac{b}{h}$
- B) $\frac{b}{k}$
- C) $\frac{45}{h}$
- D) $\frac{45}{k}$
- The expression $\frac{5x + 13}{x+2}$ can be rewritten in the form $A + \frac{B}{x+2}$, where $A$ and $B$ are constants. What is the value of $B$? (Grid-in)
- If $cx^2 + 10x + d = (ax+b)^2$ for all values of $x$, where $a,b,c,d$ are positive constants, what is the value of $c \times d$? (Grid-in)
- Which expression is equivalent to $(x^2 + y^2)^2 - (x^2 - y^2)^2$?
- A) $0$
- B) $2y^4$
- C) $4x^2y^2$
- D) $2x^4 + 2y^4$
- The expression $\frac{1}{x+3} + \frac{1}{x-3}$ is equivalent to $\frac{ax}{x^2 - 9}$ for $x \neq 3$ and $x \neq -3$. What is the value of $a$? (Grid-in)
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