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What is the expansion of (x+5)(x−7)?
x2−2x−35
x2+2x−35
x2−12x+35
x2+12x−35
x2−2x−35
Use the product of constants: 5×(−7)=−35. Only Option A and B end in −35. Then sum the constants: 5+(−7)=−2, which matches the coefficient of x in Option A.
Two binomial expressions being multiplied: (x + 5) and (x - 7).
(x+a)(x+b)=x2+(a+b)x+ab
Use the product of constants: 5×(−7)=−35. Only Option A and B end in −35. Then sum the constants: 5+(−7)=−2, which matches the coefficient of x in Option A.
Confusing the sign of the constant term: students often multiply 5×7=35 instead of 5×(−7)=−35 or calculate the sum incorrectly.
Apply Distributive Property
Distribute the terms of the first binomial (x+5) into the second (x−7) using the FOIL method.
(x⋅x)+(x⋅−7)+(5⋅x)+(5⋅−7)
Perform Multiplication
Multiply each pair of terms individually: x2, −7x, 5x, and −35.
x2−7x+5x−35
Combine Like Terms
Combine the coefficients of the middle terms: −7x+5x=−2x.
x2−2x−35
A is correct because the expansion of (x+5)(x−7) simplifies to x2−2x−35.
This is a fundamental concept for quadratic equations; mastering this is essential for factoring trinomials and solving for roots of quadratic equations.
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