Polynomials

Explanation
A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division or square roots. It can have multiple terms, with each term having a variable raised to a non-negative integer power. Therefore, any expression that involves division or square roots is not a polynomial.

Explanation
The given polynomial has three zeros: x = 0, x = -1, and x = 1. However, the only zero mentioned in the options is x = 0. Therefore, the correct answer is x = 0.

Explanation
To find the value of p(2), we substitute 2 in place of x in the given polynomial expression p(x). So, p(2) = 3(2)^6 - 5(2)^5 - 2 + 7. Simplifying this expression, we get p(2) = 3(64) - 5(32) - 2 + 7 = 192 - 160 - 2 + 7 = 37. Therefore, the correct answer is 37.

Explanation
If x = 2 is a zero of the polynomial 2x^2 + 3x - p, it means that when x is substituted with 2 in the polynomial, the resulting expression will equal zero. To find the value of p, we can substitute x with 2 in the polynomial equation and set it equal to zero. By doing so, we get 2(2)^2 + 3(2) - p = 0. Simplifying this equation gives us 8 + 6 - p = 0. Combining like terms, we have 14 - p = 0. Solving for p, we find that p = 14.

Explanation
The expression a3 + b3 - 3ab is a special case of the formula for the sum of cubes, which is (a + b)(a2 - ab + b2). In this case, since a + b = -1, we can substitute -1 for (a + b) in the formula. Simplifying further, we get (-1)(a2 - ab + b2). Since (-1)(-1) = 1, the expression simplifies to a2 - ab + b2. Plugging in the values of a and b, we get (-1)2 - (-1)(-1) + (-1)2, which simplifies to 1 + 1 + 1 = 3. Therefore, the correct answer is 1.

Explanation
When a polynomial is divided by (x + 1), we can use synthetic division to find the remainder. Plugging -1 into the polynomial, we get (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 + (-3) + 1 = 0. Since the remainder is 0, this means that (x + 1) is a factor of the polynomial, and it divides evenly with no remainder.

Explanation
The given polynomial is a cubic polynomial, which means it is of degree 3. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex roots. Since the degree of the polynomial is 3, it will have 3 zeros. Therefore, the correct answer is 3.

Explanation
The given equation, 2(a^2 + b^2) = (a + b)^2, can be simplified as 2a^2 + 2b^2 = a^2 + 2ab + b^2. By rearranging the terms, we get a^2 - 2ab + b^2 = 0, which can be factored as (a - b)^2 = 0. Taking the square root of both sides, we get a - b = 0. Adding b to both sides, we find a = b. Therefore, the correct answer is a = b.

Explanation
If (x + 2) and (x - 2) are factors of the given expression, it means that when we substitute x = -2 and x = 2 into the expression, the result is equal to zero. By substituting these values, we can determine the values of a and b. When x = -2, the expression becomes 16a - 4 + 12 - 2b - 4 = 0. Simplifying this equation gives us 16a - 2b + 4 = 0. Similarly, when x = 2, the expression becomes 16a + 4 + 12 + 2b - 4 = 0, which simplifies to 16a + 2b + 16 = 0. Solving these two equations simultaneously, we find that a = -1 and b = 0. Therefore, the value of a + b is -1 + 0 = -1.

Explanation
The given expression is a quadratic equation in the form of ax^2 + bx + c. In this case, a = 1, b = -4, and c = 1. To find the value of the expression, we can substitute the given value of x and evaluate the expression. When x = 2, the expression becomes (2^2 - 4*2 + 1) = 4 - 8 + 1 = -3 + 1 = -2. Therefore, the value of (x^2 - 4x + 1) is not equal to 3, 1, or -1, but it is equal to zero.

Explanation
When we divide the polynomial x^3 + x - 3 - 3x^2 by x - 2 using long division, we find that the remainder is 3. This means that the polynomial cannot be completely divided by x - 2, and there is a remainder of 3 left over. Therefore, the correct answer is 3.

Explanation
The zeros of a polynomial are the values of x for which the polynomial equals zero. In this case, the polynomial is x^2 - 3. To find the zeros, we set the polynomial equal to zero and solve for x. So, x^2 - 3 = 0. Adding 3 to both sides gives x^2 = 3. Taking the square root of both sides gives x = ±√3. Therefore, the zeros of the polynomial x^2 - 3 are ±√3.

Explanation
To find the remainder of the polynomial P(x) when divided by (x - 1), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial P(x) by a linear divisor (x - a) is P(a). In this case, we need to evaluate P(1):
P(1) = 3(1)^4 - 4(1)^3 + 2(1)^2 - 1 + 7
= 31 - 41 + 2*1 - 1 + 7
= 3 - 4 + 2 - 1 + 7
= 7