UpStudy Free Solution:

To find the missing polynomial, let's denote it by \(P( x) \). The given equation is:

\[P( x) - ( 20 - 4x - 5x^ 2) = 20 - 7x^ 2\]

First, we can rewrite the equation to isolate \(P( x) \):

\[P( x) = ( 20 - 7x^ 2) + ( 20 - 4x - 5x^ 2) \]

Next, combine like terms:

\[P( x) = 20 - 7x^ 2 + 20 - 4x - 5x^ 2\]

\[P( x) = ( 20 + 20) + ( - 7x^ 2 - 5x^ 2) - 4x\]

\[P( x) = 40 - 12x^ 2 - 4x\]

So, the missing polynomial is:

\[P( x) = 40 - 12x^ 2 - 4x\]

Supplemental Knowledge

Polynomials are algebraic expressions that consist of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Understanding how to manipulate and combine polynomials is fundamental in algebra.

Key Concepts:

1. Polynomial Addition and Subtraction:

- When adding or subtracting polynomials, combine like terms. Like terms have the same variable raised to the same power.

- Example: \(( 3x^ 2 + 2x + 1) + ( 4x^ 2 - x + 5) = ( 3x^ 2 + 4x^ 2) + ( 2x - x) + ( 1 + 5) = 7x^ 2 + x + 6\).

2. Combining Like Terms:

- Identify terms with the same variable and exponent.

- Combine their coefficients.

- Example: \(5x^ 3 - 3x^ 3 = ( 5- 3) x^ 3 = 2x^ 3\).

Additional Example:

Consider finding a polynomial \(Q( x) \) such that:

\[Q( x) - ( 6x^ 2 - 4x + 8) = x^ 2 + x\]

First, isolate \(Q( x) \):

\[Q( x) = ( x^ 2 + x) + ( 6x^ 2 - 4x + 8) \]

Combine like terms:

\[Q( x) = x^ 2 + x + 6x^ 2 - 4x + 8\]

\[Q( x) = ( 1+ 6) x^ 2 + ( 1- 4) x + 8\]

\[Q( x) = 7x^ 2 - 3x + 8\]

So, the polynomial \(Q( x) \):

\[Q( x) = 7x^ 2 - 3x + 8\]

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