Respuesta :
Answer:
(A) (5/2)x^3 -(9/4)x^2 +(7/2)x -3/2
(B) No. The factor 1/4x -1/2 is not the same as 1/2x -1/4.
Step-by-step explanation:
(A) With a little practice, you can do these in your head, so there is no "work" to show.
The product of highest-degree terms is (1/2x)(5x^2) = (5/2)x^3.
The x^2 term in the product will be the sum of terms that are one of ...
- x-term × x-term
- constant × x^2
so the x^2 term is ...
(1/2x)(-2x) + (-1/4)(5x^2) = -x^2 -(5/4)x^2 = -(9/4)x^2
The x-term in the product will be the sum of terms of the form
- constant × x-term
so the x term is ...
(1/2x)(6) + (-1/4)(-2x) = 3x +(1/2)x = (7/2)x
Finally, the constant term in the product is the product of the constants:
(-1/4)(6) = -6/4 = -3/2
This makes the product of the two polynomials be ...
(5/2)x^3 -(9/4)x^2 +(7/2)x -3/2
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(B) One of the factors is the same, but the other is different. The product will be different.
- 1/2x -1/4 = (2x-1)/4
- 1/4x -1/2 = (x -2)/4 . . . . not the same as above
Expressions consist of basic mathematical operators. The product of 1/2x-1/4 & 5x^2-2x+6 is not equal to the product of 1/4x-1/2 & 5x^2-2x+6.
What is an Expression?
In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.
A.) We need to find the product of the two given expressions,
[tex](\dfrac{1}{2x}-\dfrac{1}{4}) \times (5x^2-2x+6)[/tex]
In order to multiply the two given expressions, we will first multiply the first term of the first bracket with the entire expression in the second term, and then we will do the same with the second term in the first bracket.
[tex](\dfrac{1}{2x}-\dfrac{1}{4}) \times (5x^2-2x+6)\\\\=[ \dfrac{1}{2x} \times (5x^2-2x+6)] - [\dfrac{1}{4} \times (5x^2-2x+6)]\\\\= [(\dfrac{1}{2x} \times 5x^2 )-(\dfrac{1}{2x}\times 2x)+(\dfrac{1}{2x} \times 6)]- [(\dfrac{1}{4} \times 5x^2)-(\dfrac{1}{4} \times 2x)+(\dfrac{1}{4} \times 6)][/tex]
Now simplify the entire equation,
[tex]= [(\dfrac{5x}{2})-(1)+(\dfrac{3}{x})] -[(\dfrac{5x^2}{4})-(\dfrac{x}{2})+(\dfrac{3}{2})]\\\\= \dfrac{5x}{2}-1+\dfrac{3}{x} - \dfrac{5x^2}{4} + \dfrac{x}{2}+\dfrac{3}{2}\\\\\text{Rearrange the entire equation}\\\\= \dfrac{-5x}{2} + \dfrac{5x}{2}+\dfrac{x}{2}-\dfrac{3}{2}-1+\dfrac{3}{x}\\\\= \dfrac{-5x^2}{4}+\dfrac{6x}{2} - 2.5 +\dfrac{3}{x}[/tex]
Thus, the product of the two expressions is [tex]\dfrac{-5x^2}{4}+\dfrac{6x}{2} - 2.5 +\dfrac{3}{x}[/tex].
B.) To compare the two expressions we will find the value of the two given expressions,
[tex](\dfrac{1}{2x}-\dfrac{1}{4}) \times (5x^2-2x+6) = \dfrac{-5x^2}{4}+\dfrac{6x}{2} - 2.5 +\dfrac{3}{x}[/tex]
For the value of
[tex](\dfrac{1}{4x}-\dfrac{1}{2}) \times (5x^2-2x+6)[/tex]
we will simplify this,
[tex](\dfrac{1}{4x}-\dfrac{1}{2}) \times (5x^2-2x+6)\\= [\dfrac{1}{4x}\times (5x^2-2x+6)]-[\dfrac{1}{2}\times (5x^2-2x+6)]\\\\= [(\dfrac{1}{4x}\times 5x^2)-(\dfrac{1}{4x}\times 2x)+(\dfrac{1}{4x}\times 6)]-[(\dfrac{1}{2}\times 5x^2)-(\dfrac{1}{2}\times 2x)+(\dfrac{1}{2}\times 6)][/tex]
[tex]= [\dfrac{5x}{4} - \dfrac{1}{2} + \dfrac{3}{2x}]-[\dfrac{5x^2}{2} - x +3]\\\\= \dfrac{5x}{4} - \dfrac{1}{2} + \dfrac{3}{2x} - \dfrac{5x^2}{2} + x-3\\\\= \dfrac{-5x^2}{2}+ \dfrac{9x}{4}+3.5+\dfrac{3}{2x}[/tex]
As we can see the result of the product of both expressions is different.
Hence, the product of 1/2x-1/4 & 5x^2-2x+6 is not equal to the product of 1/4x-1/2 & 5x^2-2x+6.
Learn more about Expression:
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