1Cademy - Subtracting $$(p^2 + 10pq - 2q^2)$$ from $$(p^2 + q^2)$$

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Adding and Subtracting Polynomials

Example

Subtracting $(p^2 + 10pq - 2q^2)$ from $(p^2 + q^2)$

To subtract $(p^2 + 10pq - 2q^2)$ from $(p^2 + q^2)$ , first write the subtraction expression:
$(p^2 + q^2) - (p^2 + 10pq - 2q^2)$

Next, distribute the subtraction by multiplying each term in the second polynomial by $-1$ , which changes their signs:
$p^2 + q^2 - p^2 - 10pq + 2q^2$

Rearrange the terms to group the like terms together:
$p^2 - p^2 - 10pq + q^2 + 2q^2$

Finally, combine the like terms. The $p^2$ terms cancel out:
$-10pq + 3q^2$

The difference is $-10pq + 3q^2$ .

Updated 2026-05-25

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OpenStax Intermediate Algebra

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As part of developing an automated financial tracking tool, you are required to document the exact algebraic steps taken to subtract the old cost model, $(p^2 + 10pq - 2q^2)$ , from the new baseline model, $(p^2 + q^2)$ . Recall the standard procedure for polynomial subtraction and arrange the computational steps in the correct operational order.
A logistics company uses two different models to estimate operational costs. Model A is represented by the polynomial $p^2 + q^2$ , and Model B is represented by $p^2 + 10pq - 2q^2$ . If an analyst subtracts Model B from Model A to find the difference, which simplified expression represents the result?
A corporate data analyst is verifying the sign changes in a financial model used for budget forecasting. The model requires subtracting the 'Expense Projection' polynomial $(p^2 + 10pq - 2q^2)$ from the 'Revenue Baseline' polynomial $(p^2 + q^2)$ . Match each original term from the Expense Projection polynomial with its correct form after the subtraction sign has been distributed (but before combining like terms).
In a corporate financial forecasting tool, the efficiency variance is calculated by subtracting the 'Operational Risk' polynomial $(p^2 + 10pq - 2q^2)$ from the 'Planned Revenue' polynomial $(p^2 + q^2)$ . During the algebraic process of subtracting these two polynomials, the distributive property is applied to the 'Operational Risk' polynomial, which results in the $-2q^2$ term becoming positive.
An inventory analyst is auditing a database formula and must subtract the 'Shrinkage' model, $(p^2 + 10pq - 2q^2)$ , from the 'Total Stock' model, $(p^2 + q^2)$ . Recalling the proper algebraic rules for polynomial subtraction, the analyst begins by distributing the subtraction operation. This step is performed by multiplying each term in the 'Shrinkage' model by ____, which correctly changes their signs before combining like terms.

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