High School

Which is the product of [tex]\left(1+4x+3x^2\right)\left(2-7x-9x^2\right)[/tex]?

A. [tex]2-28x^2-27x^4[/tex]
B. [tex]2+x-31x^2-57x^3-27x^4[/tex]
C. [tex]2-35x-3x^2-57x^3-27x^4[/tex]
D. [tex]2-7x-3x^2-21x^3-27x^4[/tex]
E. [tex]2+15x+32x^2+57x^3+27x^4[/tex]

Answer :

Answer:

B. [tex] 2 + x - 31x^2 - 57x^3 - 27x^4 [/tex]

Step-by-step explanation:

Given:

  • To find the product of: [tex] (1 + 4x + 3x^2)(2 - 7x - 9x^2) [/tex]

Let’s expand this using the distributive property (FOIL or box method). Multiply each term from the first polynomial with each term of the second:

For 1:

  • [tex] 1 \cdot 2 = 2 [/tex]
  • [tex] 1 \cdot (-7x) = -7x [/tex]
  • [tex] 1 \cdot (-9x^2) = -9x^2 [/tex]

For 4x:

  • [tex] 4x \cdot 2 = 8x [/tex]
  • [tex] 4x \cdot (-7x) = -28x^2 [/tex]
  • [tex] 4x \cdot (-9x^2) = -36x^3 [/tex]

For 3x²:

  • [tex] 3x^2 \cdot 2 = 6x^2 [/tex]
  • [tex] 3x^2 \cdot (-7x) = -21x^3 [/tex]
  • [tex] 3x^2 \cdot (-9x^2) = -27x^4 [/tex]

Now, combine like terms:

  • Constant: [tex] 2 [/tex]
  • x terms: [tex] -7x + 8x = x [/tex]
  • x² terms: [tex] -9x^2 -28x^2 + 6x^2 = -31x^2 [/tex]
  • x³ terms: [tex] -36x^3 -21x^3 = -57x^3 [/tex]
  • x⁴ term: [tex] -27x^4 [/tex]

Therefore, the final answer results in:

[tex] \boxed{2 + x - 31x^2 - 57x^3 - 27x^4} [/tex]

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