To verify which expression is equivalent to \((x-5)(x-3)\), we can use the distributive property (also known as the FOIL method for binomials):
\[ (x - 5)(x - 3) = x \cdot x + x \cdot (-3) - 5 \cdot x - 5 \cdot (-3) \]
Calculating each part step-by-step:
- \(x \cdot x = x^2\)
- \(x \cdot (-3) = -3x\)
- \(-5 \cdot x = -5x\)
- \(-5 \cdot (-3) = 15\)
Now we can combine these parts:
\[ (x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
So, \((x-5)(x-3)\) simplifies to \(x^2 - 8x + 15\).
Now we can compare this with the provided options:
A. \(3x^2 - 4x^2 - 3x - 15\)
Simplifying A: \[ 3x^2 - 4x^2 - 3x - 15 = -x^2 - 3x - 15 \quad \text{(not equivalent)} \]
B. \(2x - 2\) (not a polynomial of the same degree; not equivalent)
C. \(x^2 - 8x + 15\) (this matches what we calculated; this is equivalent)
D. \(x^2 - 2x + 15\) (not equivalent)
The expression that is equivalent to \((x-5)(x-3)\) is:
C. \(x^2 - 8x + 15\)