Factor Expressions and Simplify Radicals

Long calculations often become short when you change their form. Your first task is to recognize what stays equal while an expression is rearranged, factored, or written with simpler radicals.

What you will learn

• Recognize identities
• Factor strategically
• Preserve equivalence

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Choose a useful representation

An identity is an equality that holds for every input for which both sides are defined. The distributive law explains factoring: multiplying a sum distributes a common factor, and factoring reverses that multiplication. For example, grouping an expression by a repeated bracket can reveal a product that is difficult to notice term by term. Before expanding, ask whether the question needs a value, a factor, or a comparison.

The difference of two squares is useful because the middle terms cancel: $(u-v)(u+v)=u^2-v^2$. A product of numbers equally far from a convenient center can therefore be calculated from the center squared minus the offset squared. This is an exact calculation, not an approximation.

Cancellation changes the appearance, not the original domain

You may cancel a common nonzero factor from a numerator and denominator. You cannot cancel a term from a sum. Factoring first makes this distinction visible: the entire numerator and denominator must each contain the factor being removed. A simplified expression can be easier to evaluate even though it retains every restriction inherited from its original denominator.

Keep a short list of forbidden values beside the calculation. If a factor disappears during simplification, its forbidden zero still matters. A formula that gives a number at that input does not repair the undefined original expression. This detail becomes especially important when a later problem asks you to solve an equation.

Radicals have multiplicative structure

For nonnegative $a,b$, the nonnegative number $\sqrt a\sqrt b$ squares to $ab$, so it equals $\sqrt{ab}$. Extracting a perfect-square factor is therefore justified. Once simplified, radicals with the same remaining square root combine by adding their coefficients. Radicals with different remaining roots generally do not.

The square of a sum still contains a cross term, even when the terms are radicals. Conjugates avoid that cross term: $(u+v)(u-v)=u^2-v^2$. Use this relationship to rationalize a denominator or evaluate a product. Always remember that $\sqrt{x^2}=|x|$ for real $x$, because the square-root symbol denotes the nonnegative root.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Canceling a term from a sum instead of a common factor.
• Forgetting a forbidden input after its denominator factor disappears.
• Treating $\sqrt{a+b}$ as $\sqrt a+\sqrt b$.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Choose a representation that matches the requested quantity.
• Use identities as justified transformations.
• Factor before canceling and preserve domain restrictions.