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Zorluk: OrtaOperations on Polynomials

Let P(x)=(3x2−2x+4)(2x−5)−(4x3−7x2+x−3)P(x) = (3x^2 - 2x + 4)(2x - 5) - (4x^3 - 7x^2 + x - 3). When P(x)P(x) is simplified and written in standard form, what is the coefficient of the x2x^2 term?

Cevap: -12

Cevap

The coefficient of the x2x^2 term is -12.
Expanding (3x2−2x+4)(2x−5)(3x^2 - 2x + 4)(2x - 5) yields 6x3−19x2+18x−206x^3 - 19x^2 + 18x - 20. Distributing the negative sign across the second polynomial yields −4x3+7x2−x+3-4x^3 + 7x^2 - x + 3. Combining the x2x^2 terms gives −19x2+7x2=−12x2-19x^2 + 7x^2 = -12x^2, so the coefficient of the x2x^2 term is -12.

Adım Adım Çözüm

1
Expand the product of the trinomial and the binomial: (3x2−2x+4)(2x−5)(3x^2 - 2x + 4)(2x - 5)
6x3−19x2+18x−206x^3 - 19x^2 + 18x - 20
To find the expanded form of the first polynomial component before subtraction
2
Distribute the negative sign across the second polynomial: −(4x3−7x2+x−3)-(4x^3 - 7x^2 + x - 3)
−4x3+7x2−x+3-4x^3 + 7x^2 - x + 3
To prepare the second polynomial for combination of like terms
3
Combine the like terms from the two expanded components
2x3−12x2+17x−172x^3 - 12x^2 + 17x - 17
To write the entire polynomial in standard form and identify the coefficient of x2x^2

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Operations on Polynomials
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Operations on Polynomials

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