High School

Which polynomial represents the sum below?

\[ (4x^2 + 3x + 4) + (-3x^6 - 4x^5 - 2x^2 + 5x) \]

A. \( 3x^6 + 4x^5 - 2x^2 - 8x + 4 \)
B. \( 3x^6 - 4x^5 + 2x^2 + 8x + 4 \)
C. \( -x^6 - 4x^5 + 2x^2 + 8x + 4 \)
D. \( -3x^6 - 4x^5 + 2x^2 + 8x + 4 \)

Answer :

The correct answer is: [D]: " -3x⁶ − 4x⁵ + 2x² + 8x + 4 " .
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Explanation:
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Given:
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" ( 4x² + 3x + 4) + ( -3x⁶ − 4x⁵ − 2x² + 5x) " ;

Rewrite as:

" ( 4x² + 3x + 4) + 1( -3x⁶ − 4x⁵ − 2x² + 5x) " ;

Note: The "1" is implied — since "1", multiplied by any value, is equal to that same value.
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Note the "distributive property of multiplication" :
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a(b + c) = ab + ac ;

a(b − c) = ab − ac ;
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So, consider the following portion of the expression:
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" + 1 ( -3x⁶ − 4x⁵ − 2x² + 5x) " ;

= (+1 * -3x⁶) − (1 * 4x⁵) − (1 * 2x²) + (1 * 5x)

= − 3x⁶ − 4x⁵ − 2x² + 5x ;

Now, bring down the rest of the expression:

( 4x² + 3x + 4) − 3x⁶ − 4x⁵ − 2x² + 5x ;

= 4x² + 3x + 4 − 3x⁶ − 4x⁵ − 2x² + 5x

Combine the "like terms" ;

+ 4x² − 2x² = + 2x² ;

+ 3x + 5x = 8x ;

And we have:

" -3x⁶ − 4x⁵ + 2x² + 8x + 4 " ;

→ which is: Answer choice: [D]: " -3x⁶ − 4x⁵ + 2x² + 8x + 4 " .
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