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NCERT Solutions · Class 9 Mathematics Introduction to Linear Polynomials

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Exercise Set 2.1 1–5 (part 1 of 8)

  1. Exercise 1

    Find the degrees of the following polynomials:
    (i)
    2x2−5x+3\displaystyle 2 x^{2}-5 x+3
    (ii)
    y3+2y−1\displaystyle y^{3}+2 y-1
    (iii)
    -9\displaystyle 9
    (iv)
    4z−3\displaystyle 4 \mathrm{z}-3

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    Read off the highest power that actually appears.
    The degree of a polynomial in one variable is the largest exponent of that variable among the terms whose coefficient is not zero.
    (i)
    In \(\displaystyle 2x^{2}-5x+3\) the powers of \(\displaystyle x\) present are \(\displaystyle x^{2}\), \(\displaystyle x^{1}\) and \(\displaystyle x^{0}\). The largest exponent is \(\displaystyle 2\).
    (ii)
    In \(\displaystyle y^{3}+2y-1\) the powers of \(\displaystyle y\) present are \(\displaystyle y^{3}\), \(\displaystyle y^{1}\) and \(\displaystyle y^{0}\). The largest exponent is \(\displaystyle 3\). (There is no \(\displaystyle y^{2}\) term, but that does not matter — the degree looks only at the highest power, not at whether every power in between appears.)
    (iii)
    \(\displaystyle -9\) contains no variable. A non-zero constant can always be written as \(\displaystyle -9 = -9 \times 1 = -9x^{0}\), so the highest power of \(\displaystyle x\) in it is \(\displaystyle 0\).
    (iv)
    In \(\displaystyle 4z-3\) the powers of \(\displaystyle z\) present are \(\displaystyle z^{1}\) and \(\displaystyle z^{0}\). The largest exponent is \(\displaystyle 1\), so this one is a linear polynomial.
    Degrees: (i) \(\displaystyle 2\) (ii) \(\displaystyle 3\) (iii) \(\displaystyle 0\) (iv) \(\displaystyle 1\)

  2. Exercise 2

    Write polynomials of degrees 1\displaystyle 1, 2\displaystyle 2 and 3.

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    Decide the highest power first; the lower terms are then free.A polynomial has degree \(\displaystyle d\) exactly when a term in \(\displaystyle x^{d}\) is present with a non-zero coefficient and no higher power appears. So write down the leading term first, then add any lower-power terms you like (or none at all).Degree $\displaystyle 1$ (a linear polynomial): \(\displaystyle 3x+4\). The highest power of \(\displaystyle x\) is \(\displaystyle 1\).Degree $\displaystyle 2$ (a quadratic polynomial): \(\displaystyle x^{2}-5x+6\). The highest power of \(\displaystyle x\) is \(\displaystyle 2\).Degree $\displaystyle 3$ (a cubic polynomial): \(\displaystyle 2x^{3}+x^{2}-7\). The highest power of \(\displaystyle x\) is \(\displaystyle 3\). Notice that the \(\displaystyle x\) term is missing here — that is perfectly allowed, because a polynomial need not contain every power below its degree.Answers may differ from person to person. Any three polynomials whose highest powers are \(\displaystyle 1\), \(\displaystyle 2\) and \(\displaystyle 3\) are equally correct — for instance \(\displaystyle y-9\), \(\displaystyle 4t^{2}+t\) and \(\displaystyle z^{3}\) form just as good a set. The only two things to check are that the leading coefficient is not zero, and that every exponent is a whole number (so \(\displaystyle \sqrt{x}+1\) and \(\displaystyle \tfrac{1}{x}\) are not polynomials).One valid answer: degree $\displaystyle 1$ — \(\displaystyle 3x+4\); degree $\displaystyle 2$ — \(\displaystyle x^{2}-5x+6\); degree $\displaystyle 3$ — \(\displaystyle 2x^{3}+x^{2}-7\).

  3. Exercise 3

    What are the coefficients of x2\displaystyle x^{2} and x3\displaystyle x^{3} in the polynomial x4−3x3+6x2−2x+7\displaystyle x^{4}-3 x^{3}+6 x^{2}-2 x+7 ?

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    A coefficient is the number multiplying that power — and the sign belongs to it.Write the polynomial with each sign shown as part of its own term: \[x^{4}+(-3)x^{3}+6x^{2}+(-2)x+7 \]Now read off the number multiplying each power. The number multiplying \(\displaystyle x^{2}\) is \(\displaystyle 6\), and the number multiplying \(\displaystyle x^{3}\) is \(\displaystyle -3\).The minus sign in front of \(\displaystyle 3x^{3}\) is part of the coefficient; it is not something separate that gets dropped. Also note that \(\displaystyle x^{4}\) has coefficient \(\displaystyle 1\), because \(\displaystyle x^{4}\) means \(\displaystyle 1 \cdot x^{4}\).Coefficient of \(\displaystyle x^{2}\) is \(\displaystyle 6\); coefficient of \(\displaystyle x^{3}\) is \(\displaystyle -3\).

  4. Exercise 4

    What is the coefficient of z\displaystyle z in the polynomial 4z3+5z2−11\displaystyle 4 z^{3}+5 z^{2}-11 ?

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    A power that is missing has coefficient \(\displaystyle 0\).The polynomial \(\displaystyle 4z^{3}+5z^{2}-11\) has a \(\displaystyle z^{3}\) term, a \(\displaystyle z^{2}\) term and a constant term — but no term in \(\displaystyle z\) itself.Putting the missing power in explicitly changes nothing, because \(\displaystyle 0 \cdot z = 0\): \[4z^{3}+5z^{2}+0 \cdot z-11 \]This is the very same polynomial, and written this way the coefficient of \(\displaystyle z\) can be read straight off. A term being absent is exactly what "its coefficient is \(\displaystyle 0\)" means — the answer is not "there is no coefficient".The coefficient of \(\displaystyle z\) is \(\displaystyle 0\).

  5. Exercise 5

    What is the constant term of the polynomial 9x3+5x2−8x−10\displaystyle 9 x^{3}+5 x^{2}-8 x-10 ? Recall that polynomials of degree 1\displaystyle 1 are called linear polynomials. In this chapter, we shall study linear polynomials.

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    The constant term is the one term that carries no variable.The polynomial \(\displaystyle 9x^{3}+5x^{2}-8x-10\) has four terms: \(\displaystyle 9x^{3}\), \(\displaystyle 5x^{2}\), \(\displaystyle -8x\) and \(\displaystyle -10\). The first three each contain \(\displaystyle x\); only the last does not, so \(\displaystyle -10\) is the constant term.A quick way to confirm it: every term containing \(\displaystyle x\) becomes \(\displaystyle 0\) when you put \(\displaystyle x=0\), so the value of the polynomial at \(\displaystyle x=0\) is precisely the constant term. \[9(0)^{3}+5(0)^{2}-8(0)-10 = 0+0-0-10 = -10 \]As always, the sign travels with the term, so the answer is \(\displaystyle -10\) and not \(\displaystyle 10\).The constant term is \(\displaystyle -10\).