Identify the polynomials among the following.

(1) x + 1

(2) $x^{2} + 1 + 2 x$

(3) $x^{3} + 1$

(4)x+ $\frac{1}{x}$

1, 2, 3

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1 and 4

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Solution

The correct option is A
1, 2, 3

Polynomial is an expression with power of the variables to be a whole numbers.

(1) x+ 1 –> Power of x is 1 => Linear polynomial

(2) $x^{2} + 1 + 2 x$ -> Powers of x are 1,2 => Quadratic polynomial

(3) $x^{3} + 1$ –> Power of x is 3 => cubic polynomial

(4) $x + \frac{1}{x}$ = $x + x^{- 1}$ -> Powers of x are -1, 1

→ –1 is not a whole number. So this expression is not a polynomial.

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Determine which of the following polynomials have $(x + 1)$ as a factor: [2 MARKS]
(i) $x^{3} + x^{2} + x + 1$
(ii) $x^{4} + x^{3} + x^{2} + x + 1$

x^{3} + x^{2} + x + 1

x^{4} + x^{3} + x^{2} + x + 1

x^{4} + 3 x^{3} + 3 x^{2} + x + 1

x^{3} - x^{2} - (2 + \sqrt{2}) x + \sqrt{2}

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