Exercise 2.1 Practice
Introduction to Polynomials
Q1: Identify Polynomials
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Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $4x^2 - 3x + 7$
(ii) $y^2 + \sqrt{2}$
(iii) $3\sqrt{t} + t\sqrt{2}$
(iv) $y + \frac{2}{y}$
(v) $x^{10} + y^3 + t^{50}$
(i) $4x^2 - 3x + 7$
(ii) $y^2 + \sqrt{2}$
(iii) $3\sqrt{t} + t\sqrt{2}$
(iv) $y + \frac{2}{y}$
(v) $x^{10} + y^3 + t^{50}$
(i) $4x^2 - 3x + 7$: Polynomial in one variable ($x$). All exponents are whole numbers.
(ii) $y^2 + \sqrt{2}$: Polynomial in one variable ($y$). Exponent is 2 (whole number).
(iii) $3\sqrt{t} + t\sqrt{2}$: Not a polynomial. $\sqrt{t} = t^{1/2}$, and 1/2 is not a whole number.
(iv) $y + \frac{2}{y}$: Not a polynomial. $\frac{2}{y} = 2y^{-1}$, and -1 is not a whole number.
(v) $x^{10} + y^3 + t^{50}$: Polynomial, but not in one variable (contains $x, y, t$).
(i) Yes, (ii) Yes, (iii) No, (iv) No, (v) No (3 variables)
Q2: Coefficients
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Write the coefficients of $x^2$ in each of the following:
(i) $2 + x^2 + x$
(ii) $2 - x^2 + x^3$
(iii) $\frac{\pi}{2} x^2 + x$
(iv) $\sqrt{2}x - 1$
(i) $2 + x^2 + x$
(ii) $2 - x^2 + x^3$
(iii) $\frac{\pi}{2} x^2 + x$
(iv) $\sqrt{2}x - 1$
(i) $2 + x^2 + x$: Coefficient of $x^2$ is 1.
(ii) $2 - x^2 + x^3$: Coefficient of $x^2$ is -1.
(iii) $\frac{\pi}{2} x^2 + x$: Coefficient of $x^2$ is $\frac{\pi}{2}$.
(iv) $\sqrt{2}x - 1$: No $x^2$ term. Coefficient is 0.
(i) 1, (ii) -1, (iii) $\pi/2$, (iv) 0
Q3: Examples
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Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Binomial of degree 35: A polynomial with two terms and highest power 35.
Example: $x^{35} + 10$.
Example: $x^{35} + 10$.
Monomial of degree 100: A polynomial with one term and highest power 100.
Example: $5y^{100}$.
Example: $5y^{100}$.
e.g., $x^{35} + 10$ and $5y^{100}$
Q4: Degree
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Write the degree of each of the following polynomials:
(i) $5x^3 + 4x^2 + 7x$
(ii) $4 - y^2$
(iii) $5t - \sqrt{7}$
(iv) 3
(i) $5x^3 + 4x^2 + 7x$
(ii) $4 - y^2$
(iii) $5t - \sqrt{7}$
(iv) 3
(i) $5x^3 + 4x^2 + 7x$: Highest power is 3. Degree = 3.
(ii) $4 - y^2$: Highest power is 2. Degree = 2.
(iii) $5t - \sqrt{7}$: Highest power is 1 ($t^1$). Degree = 1.
(iv) 3: Constant polynomial ($3x^0$). Degree = 0.
(i) 3, (ii) 2, (iii) 1, (iv) 0
Q5: Classification
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Classify the following as linear, quadratic and cubic polynomials:
(i) $x^2 + x$
(ii) $x - x^3$
(iii) $y + y^2 + 4$
(iv) $1 + x$
(v) $3t$
(vi) $r^2$
(vii) $7x^3$
(i) $x^2 + x$
(ii) $x - x^3$
(iii) $y + y^2 + 4$
(iv) $1 + x$
(v) $3t$
(vi) $r^2$
(vii) $7x^3$
Linear: Degree 1. (iv) $1+x$, (v) $3t$.
Quadratic: Degree 2. (i) $x^2+x$, (iii) $y+y^2+4$, (vi) $r^2$.
Cubic: Degree 3. (ii) $x-x^3$, (vii) $7x^3$.
Linear: (iv), (v); Quadratic: (i), (iii), (vi); Cubic: (ii), (vii)