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Zorluk: OrtaMultiplying Polynomials

The dimensions of a rectangular box are given by the expressions 5x5x, (2x−3)(2x - 3), and (x+4)(x + 4). The volume of the box is the product of its three dimensions. When the volume is expressed as a fully simplified polynomial in the form ax3+bx2+cxax^3 + bx^2 + cx, determine the value of the coefficient bb.

Cevap:The value of bb is 【25】.

Cevap

25
To find the coefficient bb, you must expand the product of the three dimensions. First, multiply the binomials using FOIL: (2x−3)(x+4)=2x2+8x−3x−12=2x2+5x−12(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12. Then, distribute 5x5x across the resulting trinomial: 5x(2x2+5x−12)=10x3+25x2−60x5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x. Comparing this to the specified polynomial form ax3+bx2+cxax^3 + bx^2 + cx, the term with x2x^2 is 25x225x^2, making the coefficient b=25b = 25.

Adım Adım Çözüm

1
Set up the volume expression by multiplying the three given dimensions.
V=5x(2x−3)(x+4)V = 5x(2x - 3)(x + 4)
The volume of a rectangular box is calculated as length ×\times width ×\times height.
2
Multiply the two binomials using the FOIL method.
(2x−3)(x+4)=2x2+8x−3x−12=2x2+5x−12(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12
It is mathematically sound and often easier to multiply the binomials first before distributing the monomial.
3
Distribute the monomial 5x5x to each term in the resulting trinomial.
5x(2x2+5x−12)=10x3+25x2−60x5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x
To fully expand the polynomial, every term inside the parentheses must be multiplied by 5x5x.
4
Identify the value of bb by matching the result to the given standard form ax3+bx2+cxax^3 + bx^2 + cx.
b=25b = 25
The variable bb represents the coefficient of the x2x^2 term in the fully expanded polynomial.

Anahtar Kavram

Multiplying Polynomials and the Distributive Property
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Konular

Multiplying Polynomials

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