Describe the details of the shape and find the area of the shaded region
A square has a length of 3x-4 and a width of 2x+2, and within the square is another rectangle with a length of x-6 and a width of x-3, the space between the two square is shaded.
The square has sides of 3x-4 and 2x+2. The area of the square is (3x-4)(2x+2) = 6x^2 - 8x + 4x - 8 = 6x^2 - 4x - 8.
The rectangle inside the square has sides of x-6 and x-3. The area of the rectangle is (x-6)(x-3) = x^2 - 3x - 6x + 18 = x^2 - 9x + 18.
To find the area of the shaded region, we need to subtract the area of the rectangle from the area of the square:
(6x^2 - 4x - 8) - (x^2 - 9x + 18)
= 6x^2 - 4x - 8 - x^2 + 9x - 18
= 5x^2 + 5x - 26
Therefore, the area of the shaded region is 5x^2 + 5x - 26.
how can it relate to Polynomials and Factoring
The area of the shaded region, as we found earlier, is represented by the polynomial expression 5x^2 + 5x - 26.
To relate this expression to polynomials and factoring, we can analyze it further. We can see that this polynomial is a quadratic trinomial, which means it has three terms with a highest degree of 2. In this case, the coefficient of x^2 is 5.
To factor this trinomial, we need to find two binomials that multiply to give us 5x^2 + 5x - 26. In other words, we are looking for two factors of 5(-26) = -130 that add up to 5.
We can search for such factors by trial and error or using different factoring techniques such as grouping, decomposition, or the AC method.
Let's try factoring the trinomial 5x^2 + 5x - 26:
5x^2 + 5x - 26
To factor this trinomial, we need to find two numbers that multiply to -130 and add up to 5. The factors of -130 are -1 and 130, -2 and 65, -5 and 26, etc.
After solving, we should find that the factored expression is:
(5x - 13)(x + 2)
Therefore, the shaded area can also be represented as (5x - 13)(x + 2), which is the factored form of the quadratic trinomial 5x^2 + 5x - 26.