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Zorluk: ZorOperations on Polynomials

A manufacturing company produces sheet metal. The area of a large rectangular sheet is represented by the expression 2x2(3x3−4x+5)2x^2(3x^3 - 4x + 5) square inches. A square piece with a side length of 3x−43x - 4 inches is cut out from the sheet. Which of the following expressions represents the area of the remaining metal sheet, in square inches, for all x>2x > 2?

  1. A
    6x6−8x3+x2+24x−166x^6 - 8x^3 + x^2 + 24x - 16
  2. B
    6x5−8x3+x2−166x^5 - 8x^3 + x^2 - 16
  3. C
    6x5−8x3+x2−24x+166x^5 - 8x^3 + x^2 - 24x + 16
  4. 6x5−8x3+x2+24x−166x^5 - 8x^3 + x^2 + 24x - 16Cevap
  5. E
    6x5−8x3+x2+24x+166x^5 - 8x^3 + x^2 + 24x + 16

Cevap

The remaining area of the metal sheet is represented by 6x5−8x3+x2+24x−166x^5 - 8x^3 + x^2 + 24x - 16.
To find the remaining area, the area of the square piece must be subtracted from the area of the large rectangular sheet. Expanding the large rectangular sheet's area yields 2x2(3x3−4x+5)=6x5−8x3+10x22x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2. Expanding the area of the square piece yields (3x−4)2=9x2−24x+16(3x - 4)^2 = 9x^2 - 24x + 16. Subtracting the second polynomial from the first requires distributing the negative sign to all terms: (6x5−8x3+10x2)−(9x2−24x+16)=6x5−8x3+10x2−9x2+24x−16(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16. Combining the quadratic terms 10x2−9x2=x210x^2 - 9x^2 = x^2 yields the correct simplified expression 6x5−8x3+x2+24x−166x^5 - 8x^3 + x^2 + 24x - 16.

Adım Adım Çözüm

1
Expand the expression for the area of the large rectangular sheet.
2x2(3x3−4x+5)=6x5−8x3+10x22x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2
Distribute the term 2x22x^2 to each term in the trinomial, using the exponent rule xa⋅xb=xa+bx^a \cdot x^b = x^{a+b} to combine the variable factors.
2
Expand the expression for the area of the square piece.
(3x−4)2=9x2−24x+16(3x - 4)^2 = 9x^2 - 24x + 16
Apply the binomial squaring formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, where a=3xa = 3x and b=4b = 4.
3
Subtract the area of the square piece from the area of the large rectangular sheet.
(6x5−8x3+10x2)−(9x2−24x+16)=6x5−8x3+10x2−9x2+24x−16(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16
Distribute the negative sign to each term of the subtracted polynomial, changing the sign of each term inside the second parenthesis.
4
Combine like terms and write the final expression in descending order.
6x5−8x3+x2+24x−166x^5 - 8x^3 + x^2 + 24x - 16
Combine the quadratic terms: 10x2−9x2=x210x^2 - 9x^2 = x^2. Arrange the remaining terms in descending order of their exponents.

Anahtar Kavram

Polynomial operations involving monomial distribution with exponent rules, binomial squaring, and polynomial subtraction with sign distribution.
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Operations on Polynomials

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