Question

what is the sum of r(x) and g(x) if $r(x) = 8x^3 + 7x^2 - 11x + 20$ and $g(x) = -5x^3 - 4x^2 - 17$?

1. $3x^3 + 11x^2 - 8$
2. $3x^3 + 3x^2 - 11x + 3$
3. $3x^3 + 7x^2 - 15x + 3$
4. $3x^6 + 3x^4 - 11x + 3$

Explanation:

Step1: Define the sum of functions

To find the sum of \( R(x) \) and \( G(x) \), we need to add the corresponding terms of the two polynomials. That is, \( (R + G)(x)=R(x)+G(x) \).

Given \( R(x) = 8x^{3}+7x^{2}-11x + 20 \) and \( G(x)=- 5x^{3}-4x^{2}-17 \), we substitute these into the sum:

\( R(x)+G(x)=(8x^{3}+7x^{2}-11x + 20)+(-5x^{3}-4x^{2}-17) \)

Step2: Combine like terms

• For the \( x^{3} \) terms: \( 8x^{3}+(-5x^{3})=(8 - 5)x^{3}=3x^{3} \)
• For the \( x^{2} \) terms: \( 7x^{2}+(-4x^{2})=(7 - 4)x^{2}=3x^{2} \)
• For the \( x \) terms: There is only \( - 11x \) in \( R(x) \) and no \( x \) term in \( G(x) \) other than what's already considered, so the \( x \) term remains \( -11x \)
• For the constant terms: \( 20+(-17)=20 - 17 = 3\)

Putting it all together, we get:

\( R(x)+G(x)=3x^{3}+3x^{2}-11x + 3 \)

Answer:

1. \( 3x^{3}+3x^{2}-11x + 3 \)