Which of the following expressions is equivalent to
?
For polynomial simplification questions, first remove all parentheses by carefully distributing any numbers or negative signs across each term inside. Then systematically combine like terms: group and add/subtract all terms, then all terms, then all constants. Pay special attention to signs when subtracting a polynomial, since that’s where most errors occur. If answer choices are given, you can also plug in a simple value like or into the original expression and each option to quickly eliminate choices that don’t match.
Hints
Notice the structure
Look at how many sets of parentheses there are and what operations connect them. Are you adding, subtracting, or multiplying expressions?
Handle the subtraction carefully
Focus on the part with the minus sign before . What happens to each term inside when you subtract a whole polynomial?
Distribute and then combine like terms
After you remove parentheses by distributing the negative sign and the 2, group together terms, terms, and constant terms separately.
Check your signs
When combining like terms, double-check the plus and minus signs, especially for the terms and the constants—small sign mistakes often change just one coefficient.
Desmos Guide
Enter the original expression
Type the original expression into Desmos as f(x) = (2x^2 - 5x + 7) - (x^2 + 4x - 3) + 2(3x - 1) so Desmos defines it as a function .
Enter each answer choice as a separate function
Type each option as its own function, for example gA(x) = x^2 + 3x + 8, gB(x) = x^2 - 3x + 4, gC(x) = 2x^2 - 3x + 8, and gD(x) = x^2 - 3x + 8.
Compare values in a table
Create a table for and one choice at a time (click the gear icon next to each function and select “table”), and plug in several -values like . The equivalent expression will be the one whose function gives exactly the same -values as for all tested -values.
Alternative: graph the differences
Define new functions like hA(x) = f(x) - gA(x) and similarly for the other choices. The correct choice will be the one whose difference graph is a horizontal line on the -axis, meaning the difference is always .
Step-by-step Explanation
Write the expression and identify the operations
Start with the original expression:
Notice there are three parts:
- The first polynomial
- A subtraction of the second polynomial
- Plus times the binomial
Our goal is to remove parentheses and then combine like terms.
Distribute the negative sign
The minus in front of means you are subtracting the entire polynomial, so distribute :
Now the expression becomes:
Be careful: every term inside the parentheses changes sign when you subtract.
Distribute the 2 in the last term
Now distribute over :
Substitute this back into the expression:
Now all parentheses are gone, and we can combine like terms.
Combine like terms to get the simplified expression
Group like terms:
- Quadratic terms: and
- Linear terms: , , and
- Constant terms: , , and
Compute each group:
- Quadratic:
- Linear:
- Constants:
Putting these together, the simplified expression is
So the equivalent expression is (choice D).