Questions: Identify the missing term that makes the following statement true. The terms of the polynomials are all written in order of decreasing degree. (4x^3+10x-6)+(□-5x-8)=4x^3-5x^2+5x-14

Identify the missing term that makes the following statement true. The terms of the polynomials are all written in order of decreasing degree. (4x^3+10x-6)+(□-5x-8)=4x^3-5x^2+5x-14
Transcript text: Identify the missing term that makes the following statement true. The terms of the polynomials are all written in order of decreasing degree. \[ \left(4 x^{3}+10 x-6\right)+(\square-5 x-8)=4 x^{3}-5 x^{2}+5 x-14 \]
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Solution

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Solution Steps

To find the missing term in the polynomial equation, we need to compare the given polynomials on both sides of the equation. The missing term is the one that, when added to the left polynomial, results in the right polynomial. We will equate the coefficients of the corresponding terms from both sides to solve for the missing term.

Step 1: Set Up the Equation

We start with the equation given in the problem: \[ (4x^{3} + 10x - 6) + (\square - 5x - 8) = 4x^{3} - 5x^{2} + 5x - 14 \] We need to find the missing term represented by \( \square \).

Step 2: Rearrange the Equation

Rearranging the equation, we can express it as: \[ 4x^{3} + 10x - 6 + \square - 5x - 8 = 4x^{3} - 5x^{2} + 5x - 14 \] This simplifies to: \[ \square + 4x^{3} + 10x - 5x - 6 - 8 = 4x^{3} - 5x^{2} + 5x - 14 \]

Step 3: Compare Coefficients

By comparing the coefficients of the polynomials on both sides, we can isolate the missing term: \[ \square + 4x^{3} + 5x - 14 = 4x^{3} - 5x^{2} + 5x - 14 \] This leads to: \[ \square = -5x^{2} \]

Final Answer

The missing term that makes the statement true is \[ \boxed{-5x^{2}} \]

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