Which of the following polynomials can approximate the expression (4x3+5)(3x6−8x2)2x2+8x−4+4x3−2x+13?(1 point)
12x7−27
6x7+15
6x7+13
6x7−7
To determine which polynomial can approximate the given expression, we need to multiply the expressions (4x^3+5) and (3x^6−8x^2), and then multiply the result by 2x^2+8x−4. Finally, add 4x^3−2x+13.
First, multiply (4x^3+5) and (3x^6−8x^2):
(4x^3+5)(3x^6−8x^2) = 12x^9−32x^5+15x^6−40x^2
Next, multiply the result by 2x^2+8x−4:
(2x^2+8x−4)(12x^9−32x^5+15x^6−40x^2) = 24x^11−64x^7+30x^8−80x^4+96x^10−256x^6+120x^7−320x^3−48x^9+128x^5−60x^6+160x^2
Finally, add 4x^3−2x+13 to the result:
24x^11−64x^7+30x^8−80x^4+96x^10−256x^6+120x^7−320x^3−48x^9+128x^5−60x^6+160x^2+4x^3−2x+13 = 24x^11−48x^9−14x^8−176x^7+68x^6−128x^5−320x^3+160x^2−2x+13
From the given options, the closest polynomial to this expression is 24x^11−48x^9−14x^8−176x^7+68x^6−128x^5−320x^3+160x^2−2x+13.
Therefore, the answer is not among the given options.
To simplify the expression (4x^3 + 5)(3x^6 - 8x^2)2x^2 + 8x - 4 + 4x^3 - 2x + 13, we need to distribute and combine like terms.
Step 1: Distribute (4x^3 + 5) to (3x^6 - 8x^2).
(4x^3 + 5)(3x^6 - 8x^2) becomes 12x^9 - 32x^5 + 15x^6 - 40x^2.
Step 2: Multiply (12x^9 - 32x^5 + 15x^6 - 40x^2) by 2x^2.
2x^2(12x^9 - 32x^5 + 15x^6 - 40x^2) becomes 24x^11 - 64x^7 + 30x^8 - 80x^4.
Step 3: Combine like terms.
24x^11 - 64x^7 + 30x^8 - 80x^4 + 8x - 4 + 4x^3 + 13 - 2x becomes 24x^11 + 30x^8 + 4x^3 - 64x^7 - 80x^4 - 2x + 8x + 13 - 4.
So, the simplified expression is 24x^11 + 30x^8 + 4x^3 - 64x^7 - 80x^4 + 6x + 9.
None of the given options match the simplified expression.