Expand and simplify:
$(3 x-3)(x+5)-(2 x+2)(3 x+3)$
We need to expand and simplify the expression \((3x-3)(x+5) - (2x+2)(3x+3)\). This involves using the distributive property, also known as the FOIL method (First, Outer, Inner, Last), to expand each pair of binomials, and then performing the subtraction to combine the results.
First, we expand the product \((3x-3)(x+5)\):
\[(3x-3)(x+5) = 3x(x) + 3x(5) - 3(x) - 3(5)\]
\[= 3x^2 + 15x - 3x - 15\]
\[= 3x^2 + 12x - 15\]
Next, we expand the product \((2x+2)(3x+3)\):
\[(2x+2)(3x+3) = 2x(3x) + 2x(3) + 2(3x) + 2(3)\]
\[= 6x^2 + 6x + 6x + 6\]
\[= 6x^2 + 12x + 6\]
Now we perform the subtraction of the two results, being very careful to distribute the negative sign across every term of the second expression:
\[(3x^2 + 12x - 15) - (6x^2 + 12x + 6)\]
\[= 3x^2 + 12x - 15 - 6x^2 - 12x - 6\]
Group the like terms (the \(x^2\) terms, the \(x\) terms, and the constants):
\[(3x^2 - 6x^2) + (12x - 12x) + (-15 - 6)\]
\[= -3x^2 + 0x - 21\]
\[= -3x^2 - 21\]
Algebraic expansion is fundamental in physics and engineering for modeling complex systems. For instance, if you are designing a trajectory for a projectile where the displacement is defined by two different velocity functions, you might need to find the difference between two areas under a curve or solve for the net acceleration by subtracting forces modeled as polynomials. Simplifying expressions allows engineers to quickly determine the resultant behavior of a system without manually calculating each component separately every time.