Equivalent Expressions

Two expressions are equivalent if they give the same value for every allowed $x$. This skill is about rewriting: expanding products, factoring polynomials, simplifying with exponent rules, and reshaping rational expressions (fractions with polynomials). Most questions show one expression and ask which choice equals it, or ask for a constant that makes two forms match.

• Expanding means multiplying every term in one factor by every term in the other, then combining like terms.
• Factoring is expanding in reverse. For $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$, then split the middle term.
• If two polynomials are equal for all $x$, their matching coefficients are equal. This lets you solve for unknown constants.
• Radicals are fractional exponents: $\sqrt[n]{x^m} = x^{m/n}$. Convert, then use exponent rules.
• To simplify a rational expression, factor top and bottom and cancel common factors. Never cancel individual terms.

• $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$.
• $a^2 - b^2 = (a + b)(a - b)$.
• Factoring by splitting: $2x^2 + 7x + 3$ has $ac = 6$; use $6$ and $1$: $2x^2 + 6x + x + 3 = (2x + 1)(x + 3)$.
• Exponent example: $\frac{x^2 \cdot \sqrt[3]{x}}{\sqrt{x^3}} = x^{2 + \frac{1}{3} - \frac{3}{2}} = x^{5/6}$.

Write $(2x - 3)(4x + 1) - 5x$ in the form $ax^2 + bx + c$.

1. Expand: $(2x)(4x) + (2x)(1) + (-3)(4x) + (-3)(1) = 8x^2 + 2x - 12x - 3$.
2. Combine: $8x^2 - 10x - 3$.
3. Subtract $5x$: $8x^2 - 15x - 3$.

Check with $x = 1$: the original is $(-1)(5) - 5 = -10$, and $8 - 15 - 3 = -10$. Correct.

The expression $\frac{3x - 4}{x + 2}$ can be rewritten as $3 + \frac{A}{x + 2}$, where $A$ is a constant. What is $A$?

1. Write the numerator in terms of the denominator: $3x - 4 = 3(x + 2) - 10$.
2. Split the fraction: $\frac{3(x + 2) - 10}{x + 2} = 3 - \frac{10}{x + 2}$.
3. So $A = -10$.

Check with $x = 0$: the original gives $\frac{-4}{2} = -2$, and $3 + \frac{-10}{2} = -2$. Correct.

• Squaring term by term. $(x + 6)^2$ is $x^2 + 12x + 36$, not $x^2 + 36$.
• Losing a negative when distributing a subtraction. $-3(x - 5) = -3x + 15$.
• Multiplying exponents that should be added. $x^2 \cdot x^3 = x^5$, not $x^6$.
• Forgetting the coefficient in a power. $(2x^5)^3 = 8x^{15}$; the $2$ is cubed too.
• Canceling terms instead of factors. In $\frac{x^2 - 16}{x^2 + 7x + 12}$, you cannot cancel the $x^2$; factor to $\frac{(x - 4)(x + 4)}{(x + 3)(x + 4)}$, then cancel $(x + 4)$.