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Zorluk: OrtaOperations on Polynomials

An industrial designer is drafting a template for a rectangular solar panel. The total area of the panel is represented by the polynomial 4x3(3x2−2x+5)4x^3(3x^2 - 2x + 5) square centimeters. A rectangular sensor cutout with an area of (2x2−3)2(2x^2 - 3)^2 square centimeters is removed from the panel. Which of the following polynomials represents the remaining area, in square centimeters, of the solar panel in terms of xx?

  1. A
    12x5−12x4+20x3−12x2+912x^5 - 12x^4 + 20x^3 - 12x^2 + 9
  2. B
    12x5−12x4+20x3−912x^5 - 12x^4 + 20x^3 - 9
  3. 12x5−12x4+20x3+12x2−912x^5 - 12x^4 + 20x^3 + 12x^2 - 9Cevap
  4. D
    12x6−4x4+12x3+12x2−912x^6 - 4x^4 + 12x^3 + 12x^2 - 9
  5. E
    12x5−12x4+20x3+12x2+912x^5 - 12x^4 + 20x^3 + 12x^2 + 9

Cevap

12x5−12x4+20x3+12x2−912x^5 - 12x^4 + 20x^3 + 12x^2 - 9
To find the remaining area, calculate the total area and subtract the cutout area. The total area is 4x3(3x2−2x+5)=12x5−8x4+20x34x^3(3x^2 - 2x + 5) = 12x^5 - 8x^4 + 20x^3. The area of the cutout is (2x2−3)2=4x4−12x2+9(2x^2 - 3)^2 = 4x^4 - 12x^2 + 9. Subtracting the cutout polynomial requires distributing the negative sign to all three terms: (12x5−8x4+20x3)−(4x4−12x2+9)=12x5−8x4+20x3−4x4+12x2−9(12x^5 - 8x^4 + 20x^3) - (4x^4 - 12x^2 + 9) = 12x^5 - 8x^4 + 20x^3 - 4x^4 + 12x^2 - 9. Combining like terms yields the expression 12x5−12x4+20x3+12x2−912x^5 - 12x^4 + 20x^3 + 12x^2 - 9.

Adım Adım Çözüm

1
Find the polynomial representing the total area of the solar panel by distributing 4x34x^3 through the expression (3x2−2x+5)(3x^2 - 2x + 5).
12x5−8x4+20x312x^5 - 8x^4 + 20x^3
When multiplying terms with the same base, add their exponents (xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}).
2
Find the polynomial representing the area of the sensor cutout by expanding (2x2−3)2(2x^2 - 3)^2.
4x4−12x2+94x^4 - 12x^2 + 9
Use the binomial squaring identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, where a=2x2a = 2x^2 and b=3b = 3.
3
Subtract the cutout area from the total area, ensuring the negative sign is distributed to every term in the cutout polynomial.
(12x5−8x4+20x3)−(4x4−12x2+9)=12x5−12x4+20x3+12x2−9(12x^5 - 8x^4 + 20x^3) - (4x^4 - 12x^2 + 9) = 12x^5 - 12x^4 + 20x^3 + 12x^2 - 9
Distributing the negative sign changes the signs of all terms in the subtracted polynomial, allowing like terms to be combined.

Anahtar Kavram

Polynomial subtraction and expansion using binomial squaring and exponent properties
Tahmini Süre:1m 30s
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Konular

Operations on Polynomials

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