Adding Two Polynomials

Add: $(4x^2 + 3x - 5) + (2x^2 - 7x + 8)$

Identify like terms: $x^2$ terms: $4x^2$ and $2x^2$ $x$ terms: $3x$ and $-7x$ Constants: $-5$ and $8$ = Three groups of like terms

Add the $x^2$ terms: $4x^2 + 2x^2 = 6x^2$ = $6x^2$

Add the $x$ terms: $3x + (-7x) = 3x - 7x = -4x$ = $-4x$

Add the constants: $-5 + 8 = 3$ = $3$

Write the final polynomial: Combine all results in standard form = $6x^2 - 4x + 3$

Answer: $6x^2 - 4x + 3$

Subtracting Polynomials

Subtract: $(5x^2 - 2x + 4) - (3x^2 + 6x - 1)$

Distribute the negative sign: Change every sign in the second polynomial: $-(3x^2 + 6x - 1) = -3x^2 - 6x + 1$ = $5x^2 - 2x + 4 - 3x^2 - 6x + 1$

Group like terms: $(5x^2 - 3x^2) + (-2x - 6x) + (4 + 1)$ = Grouped by degree

Combine $x^2$ terms: $5x^2 - 3x^2 = 2x^2$ = $2x^2$

Combine $x$ terms: $-2x - 6x = -8x$ = $-8x$

Combine constants: $4 + 1 = 5$ = $5$

Write the result: Put it all together = $2x^2 - 8x + 5$

Answer: $2x^2 - 8x + 5$

Polynomials with Missing Terms

Simplify: $(x^3 - 4x + 2) + (3x^2 + 5x - 7)$

Identify all terms by degree: First polynomial: $x^3$ (degree 3), $-4x$ (degree 1), $2$ (degree 0) Second polynomial: $3x^2$ (degree 2), $5x$ (degree 1), $-7$ (degree 0) = Some degrees have only one term

Handle terms with no match: $x^3$ has no like term - keep it $3x^2$ has no like term - keep it = $x^3 + 3x^2$

Combine $x$ terms: $-4x + 5x = x$ = $x$

Combine constants: $2 + (-7) = 2 - 7 = -5$ = $-5$

Write in standard form: Arrange by decreasing degree = $x^3 + 3x^2 + x - 5$

Answer: $x^3 + 3x^2 + x - 5$

Subtracting with Negative Leading Coefficient

Simplify: $(2x^2 - 5x + 3) - (-4x^2 + 2x - 6)$

Distribute the negative to the second polynomial: $-(-4x^2) = +4x^2$ $-(+2x) = -2x$ $-(-6) = +6$ = $2x^2 - 5x + 3 + 4x^2 - 2x + 6$

Combine $x^2$ terms: $2x^2 + 4x^2 = 6x^2$ = $6x^2$

Combine $x$ terms: $-5x - 2x = -7x$ = $-7x$

Combine constants: $3 + 6 = 9$ = $9$

Write the final answer: Standard form: highest degree first = $6x^2 - 7x + 9$

Answer: $6x^2 - 7x + 9$

Mistake: Forgetting to distribute the negative sign to ALL terms when subtracting

Why: When subtracting $(3x^2 + 2x - 1)$, the minus applies to every term inside the parentheses.

Correct: $(5x^2) - (3x^2 + 2x - 1) = 5x^2 - 3x^2 - 2x + 1$, NOT $5x^2 - 3x^2 + 2x - 1$

Mistake: Adding unlike terms together

Why: $3x^2 + 5x$ cannot be simplified because $x^2$ and $x$ are different powers.

Correct: Only combine terms with the SAME variable AND the SAME exponent. $3x^2 + 5x$ is already simplified.

Mistake: Confusing coefficients with exponents when combining

Why: When adding $3x^2 + 5x^2$, you add the coefficients (3 and 5), not the exponents.

Correct: $3x^2 + 5x^2 = 8x^2$ (NOT $8x^4$). The exponent stays the same!

Mistake: Forgetting terms with no like term

Why: If there is no matching term, the term stands alone in the answer.

Correct: $(x^3 + 2x) + (5x^2 - x) = x^3 + 5x^2 + x$. The $x^3$ and $5x^2$ have no partners.

Business: Profit Calculations

Companies calculate profit by subtracting cost from revenue, where both are often polynomial functions.

If revenue is $R(x) = 50x - 0.5x^2$ and cost is $C(x) = 20x + 100$, then profit $P(x) = R(x) - C(x) = 50x - 0.5x^2 - 20x - 100 = -0.5x^2 + 30x - 100$

Physics: Combining Motion Equations

When objects move in the same direction, their position equations can be added.

If two forces create displacements $d_1 = 3t^2 + 2t$ and $d_2 = t^2 - 5t + 10$, the total displacement is $d = d_1 + d_2 = 4t^2 - 3t + 10$

Architecture: Area Calculations

When combining or removing sections of floor plans, architects add or subtract polynomial area expressions.

A room with area $A_1 = x^2 + 6x$ has an alcove with area $A_2 = 2x + 5$ added. Total area: $A_1 + A_2 = x^2 + 8x + 5$

Q: Can I add $3x^2$ and $5x$?

A: No! These are unlike terms because they have different exponents. $3x^2 + 5x$ is already fully simplified.

Q: What happens to the exponents when I add like terms?

A: The exponents stay the same! When adding $3x^2 + 5x^2$, you only add the coefficients: $3 + 5 = 8$, so the result is $8x^2$.

Q: Why do I need to distribute the negative when subtracting?

A: Subtracting a polynomial means subtracting every term in it. The negative sign in front of the parentheses applies to all terms inside, so each sign must change.