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Zorluk: OrtaMultiplying Polynomials

Which of the following represents the fully simplified product of the polynomials (3a−2b)(3a - 2b) and (4a2+5ab−b2)(4a^2 + 5ab - b^2)?

  1. 12a3+7a2b−13ab2+2b312a^3 + 7a^2b - 13ab^2 + 2b^3Cevap
  2. B
    12a3+2b312a^3 + 2b^3
  3. C
    12a3+23a2b+7ab2−2b312a^3 + 23a^2b + 7ab^2 - 2b^3
  4. D
    12a3−6a2b2+2b312a^3 - 6a^2b^2 + 2b^3

Cevap

The fully simplified product is 12a3+7a2b−13ab2+2b312a^3 + 7a^2b - 13ab^2 + 2b^3.
The correct answer is obtained by distributing each term in the binomial to every term in the trinomial, and then combining the like terms. Multiplying 3a3a by each term yields 12a3+15a2b−3ab212a^3 + 15a^2b - 3ab^2. Multiplying −2b-2b by each term yields −8a2b−10ab2+2b3-8a^2b - 10ab^2 + 2b^3. Adding these together and combining the like terms (15a2b−8a2b=7a2b15a^2b - 8a^2b = 7a^2b and −3ab2−10ab2=−13ab2-3ab^2 - 10ab^2 = -13ab^2) results in the final expression.

Adım Adım Çözüm

1
Distribute the first term of the binomial, 3a3a, to every term in the trinomial.
3a(4a2)+3a(5ab)+3a(−b2)=12a3+15a2b−3ab23a(4a^2) + 3a(5ab) + 3a(-b^2) = 12a^3 + 15a^2b - 3ab^2
This applies the distributive property to begin multiplying the polynomials.
2
Distribute the second term of the binomial, −2b-2b, to every term in the trinomial.
−2b(4a2)−2b(5ab)−2b(−b2)=−8a2b−10ab2+2b3-2b(4a^2) - 2b(5ab) - 2b(-b^2) = -8a^2b - 10ab^2 + 2b^3
This completes the distribution process for all terms.
3
Write out the full expanded expression by combining the results from the previous steps.
12a3+15a2b−3ab2−8a2b−10ab2+2b312a^3 + 15a^2b - 3ab^2 - 8a^2b - 10ab^2 + 2b^3
This provides the complete polynomial before simplification.
4
Group and combine the like terms.
12a3+(15a2b−8a2b)+(−3ab2−10ab2)+2b3=12a3+7a2b−13ab2+2b312a^3 + (15a^2b - 8a^2b) + (-3ab^2 - 10ab^2) + 2b^3 = 12a^3 + 7a^2b - 13ab^2 + 2b^3
Combining terms with identical variable parts simplifies the expression to its final standard form.

Anahtar Kavram

Multiplying a binomial by a polynomial requires distributing every term in the first polynomial to every term in the second polynomial, carefully tracking negative signs, and then combining any like terms.
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Multiplying Polynomials

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