Let f(x) = 2x^2+5x-3, g(x) =x^2-4x+2, and h(x)=--x^2.+4x-2 Select all of the expressions below that are equivalent to 3x^2+x-1.
a) g(x) + f(x)
b) f(x) + h(x)
c) f(x) + g(x)
d) h(x) -f(x)
e) g(x) -h(x)
f) f(x) -h(x)
To determine if expressions are equivalent to 3x^2+x-1, we can compare the coefficients of each term.
The coefficient of the x^2 term in 3x^2+x-1 is 3.
The coefficient of the x term in 3x^2+x-1 is 1.
The constant term in 3x^2+x-1 is -1.
a) g(x) + f(x)
The coefficient of the x^2 term in g(x) + f(x) is 1+2 = 3 (not equivalent)
The coefficient of the x term in g(x) + f(x) is -4+5 = 1 (not equivalent)
The constant term in g(x) + f(x) is 2-3 = -1 (equivalent)
b) f(x) + h(x)
The coefficient of the x^2 term in f(x) + h(x) is 2-1 = 1 (not equivalent)
The coefficient of the x term in f(x) + h(x) is 5+4 = 9 (not equivalent)
The constant term in f(x) + h(x) is -3-2 = -5 (not equivalent)
c) f(x) + g(x)
The coefficient of the x^2 term in f(x) + g(x) is 2+1 = 3 (not equivalent)
The coefficient of the x term in f(x) + g(x) is 5-4 = 1 (not equivalent)
The constant term in f(x) + g(x) is -3+2 = -1 (equivalent)
d) h(x) - f(x)
The coefficient of the x^2 term in h(x) - f(x) is -1-2 = -3 (not equivalent)
The coefficient of the x term in h(x) - f(x) is 4-5 = -1 (equivalent)
The constant term in h(x) - f(x) is -2+3 = 1 (not equivalent)
e) g(x) - h(x)
The coefficient of the x^2 term in g(x) - h(x) is 1-(-1) = 2 (not equivalent)
The coefficient of the x term in g(x) - h(x) is -4-4 = -8 (not equivalent)
The constant term in g(x) - h(x) is 2-(-2) = 4 (not equivalent)
f) f(x) - h(x)
The coefficient of the x^2 term in f(x) - h(x) is 2-(-1) = 3 (not equivalent)
The coefficient of the x term in f(x) - h(x) is 5-4 = 1 (not equivalent)
The constant term in f(x) - h(x) is -3-(-2) = -1 (equivalent)
Therefore, the expressions that are equivalent to 3x^2+x-1 are:
a) g(x) + f(x)
c) f(x) + g(x)
f) f(x) - h(x)