A polynomial identity is an equality that holds at every input, signaled by “for all real numbers.” Choose an input that makes a factor zero to isolate one unknown, then use to group the unknowns inside the other factor. Watch the negative factor at : it turns a sum into a difference.
Hints
- Hint 1
Making a factor zero makes its whole product disappear. Solve so the unknowns inside the parentheses drop out. Which unknown remains, and how can you find its value from the other side?
- Hint 2
At , each power of equals , so becomes . The other factor is negative at that input. Write the full left side before moving terms.
- Hint 3
An identity lets you use two different inputs because it holds for every . Once you know and have an equation involving , rearrange it to find the requested sum, not just one unknown.
Step-by-step
Choose inputs that remove unknowns
Step 1Make the product disappear
The equality holds for every , so choose an input that helps. Set , giving . The whole product is then zero, leaving by itself. .
- Step 2
Find the remaining constant
Call the right-hand polynomial . Type , then . Desmos prints , so that output is .
- Step 3
Use an input that groups and
At , the first parentheses become , while becomes , not . Type ; Desmos prints . The identity now gives .
- Step 4
Rearrange for the requested sum
The equation gives a difference, so rearrange it before answering. Add to both sides: . Subtract from both sides: . Add to both sides: .
- Step 5
Calculate
Since , type . Desmos prints , so the requested sum is . Grid in 8.