The DAV Class 8 Maths Solutions and **DAV Class 8 Maths Chapter 8 Worksheet 1 **Solutions of Polynomials offer comprehensive answers to textbook questions.

## DAV Class 8 Maths Ch 8 WS 1 Solutions

Question 1.

Find out whether the given expression is a polynomial or not. If not, give reasons.

(i) 5x^{3} – 4x^{2} + \(\frac{1}{2}\)

Solution:

Yes, it is polynomial of degree 3.

(ii) √3z^{2} – 5√z + 6

Solution:

Not, it is not a polynomial as the power of z is \(\frac{1}{2}\).

(iii) 6x^{4} + \(\frac{2}{3}\) x^{3} – \(\frac{3}{4}\) x^{2} – 1

Solution:

Yes, it is a polynomial of degree 4.

(iv) 7x^{\(\frac{2}{3}\)} – 8x^{\(\frac{3}{2}\)} + x^{2}

Solution:

Not, it is not a polynomial as it contains variables with rational powers.

(v) 5x – \(\frac{1}{x}+\frac{1}{x^2}\) – 2

Solution:

Not, it is not a polynomial as power of x is an negative integer.

(vi) p^{4} – 3p^{3} – p + 1

Solution:

Yes, it is a polynomial of degree 4.

Question 2.

Write each of the following polynomials in standard form and also write down their degree:

(i) p^{6} – 8p^{9} + p^{7} + 5

Solution:

Standard form of the given polynomial is – 8p^{9} + p^{7} + p^{6} + 5

degree = 9

(ii) 4z^{3} – 3z^{5} + 2z^{4} + z + 1

Solution:

The standard form of the given polynomial is – 3z^{5} + 2z^{4} + 4z^{3} + z + 1

degree = 5

(iii) (x + \(\frac{2}{3}\)) (x + \(\frac{3}{4}\))

Solution:

(x + \(\frac{2}{3}\)) (x + \(\frac{3}{4}\)) = (x^{2})^{2} + \(\left(\frac{2}{3}+\frac{3}{4}\right)\) x + \(\frac{2}{3} \times \frac{3}{4}\)

= x^{2} + \(\frac{17}{12}\) x + \(\frac{1}{2}\), which is in standard form

degree = 2

(iv) (x^{2} – \(\frac{2}{3}\)) (x^{2} + \(\frac{4}{3}\))

Solution:

(x^{2} – \(\frac{2}{3}\)) (x^{2} + \(\frac{4}{3}\)) = (x^{2})^{2} + \(\left(-\frac{2}{3}+\frac{4}{3}\right)\) x^{2} + \(\left(-\frac{2}{3}\right)\left(\frac{4}{3}\right)\)

= x^{4} + \(\frac{2}{3}\) x^{2} – \(\frac{8}{9}\) which is in standard form.

degree = 4

(v) (z^{2} + 5) (z^{2} – 6)

Solution:

(z^{2} + 5) (z^{2} – 6) = (z^{2})^{2} + (5 – 6) z^{2} + 5 (- 6)

= z^{4} – z^{2} – 30 which is in standard form.

degree = 4

(vi) (y^{3} – 4) (y^{3} – 5)

Solution:

(y^{3} – 4) (y^{3} – 5) = (y^{3})^{2} + (- 4 – 5) y^{3} + (- 4) (- 5)

= y^{6} – 9y^{3} + 20, which is in standard form

degree = 6

(vii) (p^{2} + 2) (p^{2} + 7)

Solution:

(p^{2} + 2) (p^{2} + 7) = (p^{2})^{2} + (2 + 7) p^{2} + 2 × 7

= p^{4} + 9p^{2} + 14, which is in standard form

degree = 4

(viii) (\(\frac{5}{6}\)z – \(\frac{3}{4}\)z^{2} – \(\frac{2}{3}\)z^{3} + 1)

Solution:

Standard form of (\(\frac{5}{6}\)z – \(\frac{3}{4}\)z^{2} – \(\frac{2}{3}\) z^{3} + 1) is – \(\frac{2}{3}\) z^{3} – \(\frac{3}{4}\) z^{2} + \(\frac{5}{6}\) z + 1

degree = 3

(ix) 4p + 15p^{6} – p^{5} + 4p^{2} + 3

Solution:

Standard form of 4p + 15p^{6} – p^{5} + 4p^{2} + 3 is 15p^{6} – p^{5} + 4p^{2} + 4p + 3

degree = 6.

(x) q^{10} + q^{6} – q^{4} + q^{8}

Solution:

Standard form of q^{10} + q^{6} – q^{4} + q^{8} is q^{10} + q^{8} + q^{6} – q^{4}

degree = 10.

### DAV Class 8 Maths Chapter 8 Worksheet 1 Notes

1. Monomials: The expression that contains only one term is called a monomial.

e.g. 4x^{2}, – 9, 32mp, etc.

2. Binomials: The expression that contains two terms is called a binomial.

e.g. a + b, a + 4, lm + ms, 5 – 2xy, z^{2} – x^{2}, etc.

3. Trinomial: The expression that contains three terms is called a trinomial.

e.g. a + b + c, 2mn + 5m + n, 2a + 3x + 5z, etc.

4. Polynomial: The expression that contains many number of terms is called a polynomial.

e.g. 3x + y + 4z + 3t – 5w, 2n + 5y – 3, 4a – 5b – c – d, etc.

Example of polynomial in one variable:

P(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} + ……………….. + a_{n}x^{n}

where x is a variable and a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, …………., a_{n} are the coefficients.

**Degree of a polynomial:**

The highest power of the variable in a polynomial of one variable is called its degree.

- If degree is 1, then it is called as linear polynomial.

e.g. P(x) = ax + b - If degree is 2, then it is called as quadratic polynomial.

e.g. P(x) = ax^{2}+ bx + e - If degree is 3, then it is called as cubic polynomial.

e.g. P(x) = ax^{3}+ bx^{2}+ cx + d. - If degree is 4, then it is called as biquadratic polynomial.

e.g. P(x) = ax^{4}+ bx^{3}+ cx^{2}+ dx + e

Example 1.

Classify the following polynomials as monomials, binomials and trinomials:

(i) x + y

(ii) 100

(iii) x^{2} + x^{3} + 1

(iv) 2y – 3y^{2} + 4y^{3}

(v) ab + bc + cd

(vi) 3x

Solution:

(i) binomial

(ii) monomial

(iii) trinomial

(iv) trinomial

(v) trinomial

(vi) monomial

Example 2.

Write the following polynomials in standard form:

(i) – √3x + 5 – 2√2x^{3} + 5x^{2} – x^{4}

(ii) 5 + 3x – 4x^{2} + 7x^{3} + x^{6}

(iii) – 6 + x^{9} – 3x^{8} – 2x^{3}

(iv) 3x – 5x^{2} + 7x^{3} + 8x^{5}.

(v) x^{5} – x^{4} + 3x^{3} + x + 2.

Solution:

(i) – √3x + 5 – 2√2x^{3} + 5x^{2} – x^{4}

Standard form is: – x^{4} – 2√2x^{3} + 5x^{2} – √3x + 5

(ii) 5 + 3x – 4x^{2} + 7x^{3} + x^{6}

Standard form is: x^{6} + 7x^{3} – 4x^{2} + 3x + 5

(iii) – 6 + x^{9} – 3x^{8} – 2x^{3}

Standard form is: x^{9} – 3x^{8} – 2x^{3} – 6

(iv) 3x – 5x^{2} + 7x^{3} + 8x^{5}

Standard form is: 8x^{5} + 7x^{3} – 5x^{2} + 3x

(v) x^{5} – x^{4} + 3x^{3} + x + 2

Standard form is: x^{5} – x^{4} + 3x^{3} + x + 2

Example 3:

Divide 8x^{3} – 4x^{2} – 2x by 4x.

Solution:

8x^{3} – 4x^{2} – 2x ÷ 4x

= \(\frac{8 x^3-4 x^2-2 x}{4 x}\)

= \(\frac{8 x^3}{4 x}-\frac{4 x^2}{4 x}-\frac{2 x}{4 x}\)

= 2x^{2} – x – \(\frac{1}{2}\)

Example 4.

Divide 8y^{3} + 10y – 6y^{2} + 3 by 1 + 4y by long division method and verify the

answer.

Solution:

Let us write the dividend and the divisor in standard form:

8y^{3} – 6y^{2} + 10y + 3 and 4y + 1

Verification:

Remainder = 0

∴ Dividend = Divisor × Quotient

8y^{3} – 6y^{2} + 10y + 3 = (4y + 1) × (2y^{2} – 2y + 3)

= 4y (2y^{2} – 2y + 3) + 1 (2y^{2} – 2y + 3)

= 8y^{3} – 8y^{2} + 12y + 2y^{2} – 2y + 3

= 8y^{3} – 6y^{2} + 10y + 3