Signs and Fractions

The Sign Trap

Look at this subtraction: x squared minus the quantity three x minus five. A common mistake is to write x squared minus three x minus five. This sign error quietly costs points on calculus problems. To do it correctly, you must treat the negative sign as a negative one that distributes to every term inside. The three x becomes negative, but the negative 5 flips to a positive 5.

The Rule of Distribution

Think of parentheses as a gate. When a negative sign sits outside, it flips the sign of every single term as it enters. Negative the quantity a plus b becomes negative a minus b. And negative the quantity a minus b becomes negative a plus b. In calculus, you'll see this most often when subtracting an entire function, f of x.

Practice: Distribute the Sign

Great work. The final expression is negative x squared plus four x minus seven. Try it yourself. Click each term inside the parentheses to apply the negative sign and find the simplified form. Correct. The x squared becomes negative. Exactly. The negative four x flips to positive four x. Right. The positive 7 becomes negative 7.

Subtracting a Function

If f of x is x squared plus two x, how do we calculate ten minus f of x? Select the version that correctly handles the subtraction. Not quite. Without parentheses, you only subtracted the first part of the function. Remember to subtract the whole thing. Correct! You must wrap the function in parentheses first: ten minus the quantity x squared plus two x, then distribute to get ten minus x squared minus two x.

The Complex Fraction

This looks intimidating, but we can break it down. Step one: Find a common denominator for the top fractions. One over x and one over three become the quantity three minus x, over three x. Step two: Rewrite the big fraction. Step three: Multiply by the reciprocal of the bottom. Dividing by the quantity x minus three is the same as multiplying by one over the quantity x minus three. Finally, notice that three minus x is just negative the quantity x minus three. They cancel out, leaving us with negative one over three x.

The Fraction Strategy

To clear 'fractions within fractions,' follow two rules. First, combine the top into a single fraction. Second, multiply by the reciprocal of the bottom term. Remember, A over B, all over C, is just A over B times one over C.

Practice: Dividing Fractions

Simplify this expression: one over x, divided by x. Is the answer 1, or is it one over x squared? Not quite. Remember the reciprocal rule. You are multiplying one over x by one over x, not by x. Exactly. One over x divided by x is one over x times one over x, which equals one over x squared. A common mistake is thinking the x's cancel out.

The Calculus Challenge

This is a real calculus problem: the difference quotient for 1 over x. The calculus part is done. Your job is to simplify the algebra. Start with a common denominator, x times the quantity x plus h. Type the final simplified expression in the box.

Takeaways: Signs & Fractions

Let's recap. Never subtract a multi-term expression without parentheses. Always distribute the negative to every single term inside. And for complex fractions, simplify the top before you flip the bottom. Hover over the terms on the right to refresh your memory on the basics.