Class Ix Polynomial Test

Explanation
The given expression is a quadratic trinomial. To factorize it, we can look for two binomials that multiply together to give the trinomial. In this case, the factors are (3x + 1) and (2x + 3). When we expand these binomials using the distributive property, we get 6x^2 + 11x + 3, which matches the original expression. Therefore, the correct factorization is 3x + 1) (2x + 3).

Explanation
When multiplying square roots, you can simply multiply the numbers inside the square roots and keep the square root symbol. In this case, √12 multiplied by √15 equals √180. Simplifying further, you can break down 180 into its prime factors: 2 x 2 x 3 x 3 x 5. Taking out pairs of the same number, we are left with 2 x 3 x √5, which can be simplified to 6√5. Therefore, the correct answer is 6√5.

Explanation
When we substitute x = -2 into the given polynomial, we get 7(-2)^4 + 3(-2)^2 - 4. Simplifying this expression, we have 7(16) + 3(4) - 4, which equals 112 + 12 - 4 = 120. Therefore, the value of the polynomial when x = -2 is 120.

Explanation
The degree of a polynomial is defined as the highest power of the variable in the polynomial. However, a zero polynomial is a polynomial in which all the coefficients are zero. Since there are no terms with non-zero coefficients in a zero polynomial, it does not have a highest power or degree. Therefore, the degree of a zero polynomial is not defined.

Explanation
If x + 1 is a factor of the polynomial 2x^2 + kx, it means that when x = -1, the polynomial will equal to zero. Substituting -1 into the polynomial, we get 2(-1)^2 + k(-1) = 0. Simplifying this equation, we get 2 - k = 0, which means k = 2. Therefore, the value of k is 2.

Explanation
When p + q + r = 0, it means that the sum of p, q, and r is equal to zero. This can be rearranged as p = -q - r. When we substitute this value of p into the expression p^3 + q^3 + r^3, we get (-q - r)^3 + q^3 + r^3. Simplifying this expression, we get -3q^2r - 3qr^2. Factoring out a -3qr from this expression, we get -3qr(q + r). Since q + r is equal to -p, we can further simplify the expression to -3qr(-p). This is equal to 3pqr, which can be written as 3abc. Therefore, the correct answer is 3abc.