Q.1
Let ℝ denote the set of all real numbers. Let a_i, b_i \in \mathbb{R} for i \in \{1, 2, 3\}.
Define the functions f: \mathbb{R} \to \mathbb{R}, g: \mathbb{R} \to \mathbb{R}, and h: \mathbb{R} \to \mathbb{R} by
f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4,
g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4,
h(x) = f(x+1) - g(x+2).
If f(x) \ne g(x) for every x \in \mathbb{R}, then the coefficient of x^3 in h(x) is:
Options:
(A) 8
(B) 2
(C) -4
(D) -6
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In this problem, we need to find the coefficient of x cubed in the function h(x). We're given that f(x) equals a_1 plus 10x plus a_2 x squared plus a_3 x cubed plus x to the fourth power. And g(x) equals b_1 plus 3x plus b_2 x squared plus b_3 x cubed plus x to the fourth power. The function h(x) is defined as f(x+1) minus g(x+2). We're also told that f(x) is not equal to g(x) for every real number x. Let's find the coefficient of x cubed in h(x).
Let's start by expanding f(x+1). We substitute x+1 for x in the function f. So f(x+1) equals a_1 plus 10 times (x+1) plus a_2 times (x+1) squared plus a_3 times (x+1) cubed plus (x+1) to the fourth power. Now we need to expand each term. 10 times (x+1) equals 10x plus 10. a_2 times (x+1) squared equals a_2 times (x squared plus 2x plus 1), which equals a_2 x squared plus 2a_2 x plus a_2. a_3 times (x+1) cubed equals a_3 times (x cubed plus 3x squared plus 3x plus 1), which equals a_3 x cubed plus 3a_3 x squared plus 3a_3 x plus a_3. (x+1) to the fourth power equals x to the fourth plus 4x cubed plus 6x squared plus 4x plus 1. Collecting the terms, we find that the coefficient of x cubed in f(x+1) is a_3 plus 4.
Now let's expand g(x+2). We substitute x+2 for x in the function g. So g(x+2) equals b_1 plus 3 times (x+2) plus b_2 times (x+2) squared plus b_3 times (x+2) cubed plus (x+2) to the fourth power. Let's expand each term. 3 times (x+2) equals 3x plus 6. b_2 times (x+2) squared equals b_2 times (x squared plus 4x plus 4), which equals b_2 x squared plus 4b_2 x plus 4b_2. b_3 times (x+2) cubed equals b_3 times (x cubed plus 6x squared plus 12x plus 8), which equals b_3 x cubed plus 6b_3 x squared plus 12b_3 x plus 8b_3. (x+2) to the fourth power equals x to the fourth plus 8x cubed plus 24x squared plus 32x plus 16. Collecting the terms, we find that the coefficient of x cubed in g(x+2) is b_3 plus 8.
Now we can find the coefficient of x cubed in h(x). Recall that h(x) equals f(x+1) minus g(x+2). From our previous calculations, we found that the coefficient of x cubed in f(x+1) is a_3 plus 4, and the coefficient of x cubed in g(x+2) is b_3 plus 8. Therefore, the coefficient of x cubed in h(x) equals (a_3 plus 4) minus (b_3 plus 8), which simplifies to a_3 minus b_3 minus 4. To determine the final answer, we need to find the value of a_3 minus b_3.
To find the value of a_3 minus b_3, we use the condition that f(x) is not equal to g(x) for all real values of x. This means that f(x) minus g(x) is never equal to zero. Let's expand f(x) minus g(x). We get (a_1 minus b_1) plus 7x plus (a_2 minus b_2)x squared plus (a_3 minus b_3)x cubed. Now, for a polynomial to have no real roots, it must have even degree. If a_3 is not equal to b_3, then f(x) minus g(x) would be a cubic polynomial, which always has at least one real root. This contradicts our condition. Therefore, a_3 must equal b_3, which means a_3 minus b_3 equals zero. Substituting this into our expression for the coefficient of x cubed in h(x), we get a_3 minus b_3 minus 4, which equals 0 minus 4, which equals negative 4. Therefore, the coefficient of x cubed in h(x) is negative 4, and the answer is option (C).