Question

Write the correct alternative answer for each of the following questions.

(i) Which of the following is a polynomial ?
(A) $\frac{x}{y}$ (B) $x^{\sqrt{2}}-3x$ (C) $x^{-2}+7$ (D) $\sqrt{2}{x}^2+\frac{1}{2}$

(ii) What is the degree of the polynomial $\sqrt{7}$?
(A) $\frac{1}{2}$ (B) 5 (C) 2 (D) 0

(iii) What is the degree of the 0 polynomial ?

(A) 0 (B) 1 (C) undefined (D) any real number

(iv) What is the degree of the polynomial 2x² + 5x³ + 7?
(A) 3 (B) 2 (C) 5 (D) 7

(v) What is the coefficient form of $x^3-1$ ?
(A) (1, $-$ 1) (B) (3,$-$1) (C) (1, 0, 0, $-$1) (D) (1, 3, $-$ 1)

(vi) $p\left(x\right)=x^2-7\sqrt{7}x+3$ then $p\left(7\sqrt{7}\right)=?$
(A) 3 (B) $7\sqrt{7}$ (C) $42\sqrt{7}+3$ (D) $49\sqrt{7}$

(vii) When $x=-1$ , what is the value of the polynomial 2x³ + 2x ?
(A) 4 (B) 2 (C) $-2$ (D)$-4$

(viii) If $x-1$, what is a factor of the polynomial $3x^2+mx$ then find the value of m.
(A) 2 (B) $-$2 (C) $-3$ (D) 3

(ix) Multiply ( x²$-$3) (2x $-$ 7x ³ + 4) and write the degree of the product.
(A) 5 (B) 3 (C) 2 (D) 0

(x) Which of the following is a linear polynomial ?

(A) x + 5

(B) x² + 5

(C) x³ + 5

(D) x⁴ + 5

Answer 1

(i)
In an algebraic expression, if the powers of the variables are whole numbers then the algebraic expression is a polynomial.

In the expression $\sqrt{2}{x}^2+\frac{1}{2}$, the power of the variable x is 2, which is a whole number. So, the expression $\sqrt{2}{x}^2+\frac{1}{2}$ is a polynomial.

Hence, the correct answer is option (D).

(ii)
$\sqrt{7}=\sqrt{7}\times 1=\sqrt{7}{x}^0$

The degree of the polynomial $\sqrt{7}$ is 0.

Hence, the correct answer is option (D).

(iii)
The degree of 0 polynomial is not defined.

Hence, the correct answer is option (C).

(iv)
The highest power of the variable in a polynomial is called the degree of the polynomial.

The degree of the polynomial 2x² + 5x³ + 7 is 3.

Hence, the correct answer is option (A).

(v)
$x^3-1=x^3+0x^2+0x-1$

The coefficient form of the polynomial$x^3-1$ is (1, 0, 0, −1).

Hence, the correct answer is option (C).

(vi)
$p\left(x\right)=x^2-7\sqrt{7}x+3$

$\therefore p\left(7\sqrt{7}\right)={\left(7\sqrt{7}\right)}^2-7\sqrt{7}\times 7\sqrt{7}+3=343-343+3=3$

Hence, the correct answer is option (A).

(vii)
Let p(x) = 2x³ + 2x

∴ p(−1) = 2 × (−1)³ + 2 × (−1) = 2 × (−1) − 2 = −2 − 2 = −4

Thus, the value of the polynomial when x = −1 is −4.

Hence, the correct answer is option (D).

(viii)
Let p(x) = 3x² + mx.

(x − 1) is a factor of p(x).

∴ p(1) = 0

⇒ 3 × (1)² + m × 1 = 0

⇒ 3 + m = 0

⇒ m = −3

Hence, the correct answer is option (C).

(ix)
$$\begin{gathered} \left(x^2-3\right)\left(2x-7x^3+4\right) \\ =x^2\left(2x-7x^3+4\right)-3\left(2x-7x^3+4\right) \\ =2x^3-7x^5+4x^2-6x+21x^3-12 \\ =-7x^5+2x^3+21x^3+4x^2-6x-12 \\ =-7x^5+23x^3+4x^2-6x-12 \end{gathered}$$
Thus, the degree of the resultant polynomial is 5.

Hence, the correct answer is option (A).

(x)
A polynomial with degree one is called a linear polynomial.

Thus, the polynomial x + 5 is a linear polynomial.

Hence, the correct answer is option (A).