Equivalent expressions

Most are multiple choice. The student-produced response (SPR) versions ask for one constant, such as a coefficient, a remainder or the value of $a + b$.

Core ideas

Polynomial operations

• Subtracting a polynomial changes the sign of every term inside the parentheses: $-(x^2 - 6x + 1) = -x^2 + 6x - 1$.
• Multiplying means multiplying every term by every term. For two binomials, use FOIL. For longer factors, distribute term by term, then combine like terms.
• Often you need only one coefficient of the product. The $x$-term of $(x + 4)(3x - 5)$ is $-5x + 12x = 7x$.

Factoring patterns

A linear factor $px - q$ divides a polynomial exactly when $x = \frac{q}{p}$ makes the polynomial equal zero. This gives a fast way to test factor choices.

Exponent rules

To convert a mixed exponent into radical form, split off the whole-number part: $y^{5/2} = y^2 \cdot y^{1/2} = y^2\sqrt{y}$, and $x^{4/3} = x\sqrt[3]{x}$.

Rational expressions

• To add or subtract, use a common denominator. Multiply each numerator by the factors its denominator is missing.
• To simplify, factor, then cancel common factors only. In $\frac{x + 3}{x}$, nothing cancels.
• To divide polynomials, write $\frac{p(x)}{x - a} = q(x) + \frac{r}{x - a}$. By the remainder theorem, $r = p(a)$.
• For a complex fraction, combine the inner fractions into a single fraction, then take the reciprocal. The reciprocal of a sum is not the sum of the reciprocals.

“True for all $x$” means match coefficients

If two polynomials are equal for every $x$, then the coefficients of each power of $x$ are equal. Expand, line up the $x^3$, $x^2$, $x$ and constant terms, and solve the simplest equations first. The same idea works for fractions: $\frac{ax - 10}{2x - 5}$ is constant exactly when the numerator is a multiple of the denominator (here $a = 4$).

Rearranging formulas

Undo the operations on the target variable in reverse order. If the target sits inside a reciprocal, isolate the reciprocal, combine the other side into a single fraction, and flip both sides at the end. If the target is squared, take the square root last (the positive root for lengths and other positive quantities).

Worked examples

Exam technique

• Plug in a number. Pick a simple value such as $x = 2$, evaluate the original and every choice, and keep the one that matches. Avoid 0 and 1, which can make different expressions look equal. If two choices tie, try a second value.
• Graph it in Desmos. Enter the original expression as $y = \ldots$, then each choice. The equivalent choice lies exactly on top of the original. A table of values also works for comparing outputs.
• Match the easy coefficients first. The leading coefficient and the constant term take seconds. Use the middle terms last, as a check.
• Convert radicals to exponents before simplifying, then convert back to match the answer choices.
• For factor questions, test zeros. $5x + 2$ is a factor exactly when the expression equals zero at $x = -\frac{2}{5}$. You can also graph the expression and read its $x$-intercepts.
• Sense-check SPR answers. A coefficient or remainder is usually an integer or a simple fraction. If you get a strange decimal, recheck your expansion.

Common mistakes

Quick recap

• Subtraction changes the sign of every term in the parentheses.
• Know the factoring patterns: GCF, difference of squares, perfect squares and trinomials (AC method).
• Exponents add when you multiply, subtract when you divide and multiply for a power of a power; $x^{m/n} = \sqrt[n]{x^m}$.
• Rational expressions need a common denominator. Cancel only factors. The remainder on division by $x - a$ is $p(a)$.
• “True for all $x$” means match coefficients, starting with the leading and constant terms.
• Rearrange formulas by undoing operations in reverse order, and combine fractions before taking reciprocals.
• When stuck, plug in a number or graph the original and the choices in Desmos.