Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $(- 6 x^{3} + 7 x^{2} - 9 x + 3) + (14 x^{3} + 3 x^{2} - 11 x - 7)$

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Identify the polynomials to be added: $(-6x^3 + 7x^2 - 9x + 3)$ and $(14x^3 + 3x^2 - 11x - 7)$.

Combine like terms by adding the coefficients of corresponding powers of $x$: for $x^3$, $x^2$, $x$, and the constant terms separately.

Write the sum of the $x^3$ terms: $(-6x^3) + (14x^3) = (-6 + 14)x^3$.

Write the sum of the $x^2$ terms: $(7x^2) + (3x^2) = (7 + 3)x^2$, the sum of the $x$ terms: $(-9x) + (-11x) = (-9 - 11)x$, and the sum of the constants: $3 + (-7) = 3 - 7$.

Express the resulting polynomial in standard form by arranging terms from highest to lowest degree, then identify the degree as the highest power of $x$ with a nonzero coefficient.

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주요 개념

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Polynomial Addition

Polynomial addition involves combining like terms, which are terms with the same variable raised to the same power. To add polynomials, align terms by degree and add their coefficients. This process simplifies the expression into a single polynomial.

Standard Form of a Polynomial

The standard form of a polynomial arranges terms in descending order of their degrees, starting with the highest power of the variable. Writing polynomials in standard form helps clearly identify the degree and simplifies further operations.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the expression after simplification. It indicates the polynomial's order and is important for understanding its behavior and graph.

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (−6x3+7x2−9x+3)+(14x3+3x2−11x−7)(-6x^3+7x^2-9x+3)+(14x^3+3x^2-11x-7)