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How to Simplify the Sum of Two Univariate Polynomials: $(-2v^{2}-3v+6)+(-4v^{2}+7)$
Mathematics
Grade 9 (Junior High School)
Question Content
Simplify the sum of the two univariate polynomials: $(-2v^{2}-3v+6)+(-4v^{2}+7)$
Correct Answer
$-6v^2 - 3v + 13$
Detailed Solution Steps
1
Step 1: Remove the parentheses. Since the second polynomial is added, the signs of its terms remain unchanged: $-2v^{2}-3v+6-4v^{2}+7$
2
Step 2: Group like terms together. Combine the quadratic terms, linear terms, and constant terms separately: $(-2v^2 - 4v^2) + (-3v) + (6 + 7)$
3
Step 3: Combine the coefficients of like terms. For quadratic terms: $-2 - 4 = -6$, so we get $-6v^2$. The linear term $-3v$ has no like terms, so it stays as is. For constant terms: $6 + 7 = 13$.
4
Step 4: Combine all the simplified terms to get the final result: $-6v^2 - 3v + 13$
Knowledge Points Involved
1
Like Terms
Like terms are terms in a polynomial that have the same variables raised to the same powers. For example, $-2v^2$ and $-4v^2$ are like terms, as are the constant terms 6 and 7. Only like terms can be combined through addition or subtraction.
2
Adding Polynomials
To add two polynomials, first remove any parentheses (preserving the sign of each term if the polynomial is added), then group and combine like terms. This process simplifies the polynomial to its standard form, ordered by descending powers of the variable.
3
Standard Form of a Univariate Polynomial
A univariate polynomial is written in standard form when its terms are ordered from the highest degree term to the lowest degree term. For the result $-6v^2 - 3v + 13$, the quadratic term (degree 2) comes first, followed by the linear term (degree 1), then the constant term (degree 0).
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