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Math Notes For Class 8 Chapter 3 Ex 3.1 Solved & Quiz
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Math Notes For Class 8 Chapter 3 Ex 3.1 Solved & Quiz
In 8th class maths, understanding polynomials forms the structural backbone of high school algebra. An algebraic expression consists of constants, variables, and algebraic operations (addition, subtraction, multiplication, and division).
| Concept / Operation | Mathematical Rule / Definition | Example |
|---|---|---|
| Standard Form | Polynomial terms written in descending powers of variable $x$. | $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ |
| Exponent Addition Law | When multiplying like variables, powers add: $x^a \cdot x^b = x^{a+b}$ | $2x^2 \cdot 3x^3 = 6x^{2+3} = 6x^5$ |
| Polynomial Subtraction Rule | $P(x) - Q(x) = P(x) + [-Q(x)]$. Change signs of all terms of $Q(x)$. | $(5x^2) - (-2x^2 + 3) = 5x^2 + 2x^2 - 3 = 7x^2 - 3$ |
| Degree Rule (Multivariable) | $\text{Degree of } x^a y^b z^c = a + b + c$ | $\text{Degree of } 4x^3 y^2 = 3 + 2 = 5$ |
Question 1 (Part i): Identify whether the algebraic expression $P(x) = 4x^3 - 7x^2 + 2x - 9$ is a polynomial. If yes, state its degree, leading coefficient, and constant term.
Solution:
1. Examine the exponents of variable $x$: powers are $3, 2, 1, 0$. All exponents are non-negative integers.
2. Therefore, $P(x)$ is a polynomial.
3. The highest power of $x$ is $3$, so the degree is $3$.
4. The term with the highest power is $4x^3$, so the leading coefficient is $4$.
5. The term without a variable is $-9$.
Answer: $$\text{It is a Polynomial; Degree} = 3, \text{Leading Coefficient} = 4, \text{Constant Term} = -9$$
Question 1 (Part ii): Identify whether the algebraic expression $Q(x) = 3x^2 + 5\sqrt{x} - 2$ is a polynomial. If yes, state its degree, leading coefficient, and constant term.
Solution:
1. Rewrite radical terms in exponent form: $5\sqrt{x} = 5x^{1/2}$.
2. The exponent of $x$ in the second term is $\frac{1}{2}$, which is not a whole number (non-negative integer).
3. Therefore, $Q(x)$ is not a polynomial.
Answer: $$\text{Not a polynomial (due to fractional power } x^{1/2}\text{)}$$
Question 1 (Part iii): Identify whether the algebraic expression $R(x, y) = 5x^2y^3 - 2xy^2 + 8y - 11$ is a polynomial. If yes, state its degree, leading coefficient, and constant term.
Solution:
1. Check exponents of variables $x$ and $y$: term powers are $(2,3)$, $(1,2)$, $(0,1)$, $(0,0)$. All are non-negative integers.
2. Thus, $R(x, y)$ is a polynomial.
3. Calculate the degree of each term:
- Term $5x^2y^3$: degree $= 2 + 3 = 5$
- Term $-2xy^2$: degree $= 1 + 2 = 3$
- Term $8y$: degree $= 1$
- Term $-11$: degree $= 0$
4. The maximum degree among all terms is $5$.
5. The term with the highest degree is $5x^2y^3$, so its coefficient is $5$.
6. The constant term is $-11$.
Answer: $$\text{It is a Polynomial; Degree} = 5, \text{Leading Coefficient} = 5, \text{Constant Term} = -11$$
Question 2 (Part i): Arrange the polynomial $3x + 5x^3 - 2 + 7x^2$ in descending order of powers of $x$.
Solution:
1. Identify the exponents of $x$ for each term:
- $5x^3$ has power $3$
- $7x^2$ has power $2$
- $3x$ has power $1$
- $-2$ has power $0$
2. Arrange terms from highest exponent to lowest exponent:
$$5x^3 + 7x^2 + 3x - 2$$
Answer: $$5x^3 + 7x^2 + 3x - 2$$
Question 2 (Part ii): Arrange the polynomial $8 - x^4 + 2x^2 - 6x^3 + 5x$ in descending order of powers of $x$.
Solution:
1. Identify terms with their powers of $x$:
- $-x^4$ (power 4)
- $-6x^3$ (power 3)
- $+2x^2$ (power 2)
- $+5x$ (power 1)
- $+8$ (power 0)
2. Ordering from highest to lowest degree yields:
$$-x^4 - 6x^3 + 2x^2 + 5x + 8$$
Answer: $$-x^4 - 6x^3 + 2x^2 + 5x + 8$$
Question 2 (Part iii): Arrange the polynomial $x^2y^2 - 4x^3y + 9x - 2y^3$ in descending order of powers of variable $x$.
Solution:
1. Identify powers of $x$ in each term:
- $-4x^3y$ has $x^3$ (power 3)
- $x^2y^2$ has $x^2$ (power 2)
- $9x$ has $x^1$ (power 1)
- $-2y^3$ has $x^0$ (power 0)
2. Ordering by powers of $x$ descending:
$$-4x^3y + x^2y^2 + 9x - 2y^3$$
Answer: $$-4x^3y + x^2y^2 + 9x - 2y^3$$
Question 3 (Part i): Add the polynomials $P(x) = 5x^3 - 3x^2 + 4x - 7$ and $Q(x) = 2x^3 + 8x^2 - 9x + 3$.
Solution:
1. Write the addition expression:
$$P(x) + Q(x) = (5x^3 - 3x^2 + 4x - 7) + (2x^3 + 8x^2 - 9x + 3)$$
2. Group like terms together:
$$= (5x^3 + 2x^3) + (-3x^2 + 8x^2) + (4x - 9x) + (-7 + 3)$$
3. Combine numerical coefficients:
$$= (5+2)x^3 + (-3+8)x^2 + (4-9)x + (-7+3)$$
$$= 7x^3 + 5x^2 - 5x - 4$$
Answer: $$7x^3 + 5x^2 - 5x - 4$$
Question 3 (Part ii): Add the polynomials $P(a, b) = 3a^2 - 4ab + 2b^2$ and $Q(a, b) = 7a^2 + 6ab - 5b^2$.
Solution:
1. Set up addition equation:
$$P(a,b) + Q(a,b) = (3a^2 - 4ab + 2b^2) + (7a^2 + 6ab - 5b^2)$$
2. Group like terms:
$$= (3a^2 + 7a^2) + (-4ab + 6ab) + (2b^2 - 5b^2)$$
3. Simplify terms:
$$= (3+7)a^2 + (-4+6)ab + (2-5)b^2$$
$$= 10a^2 + 2ab - 3b^2$$
Answer: $$10a^2 + 2ab - 3b^2$$
Question 4 (Part i): Subtract polynomial $Q(x) = 2x^3 - 9x^2 - 3x + 8$ from $P(x) = 6x^3 - 4x^2 + 7x - 2$.
Solution:
1. Write the subtraction expression $P(x) - Q(x)$:
$$P(x) - Q(x) = (6x^3 - 4x^2 + 7x - 2) - (2x^3 - 9x^2 - 3x + 8)$$
2. Distribute the negative sign to all terms inside the second bracket:
$$= 6x^3 - 4x^2 + 7x - 2 - 2x^3 + 9x^2 + 3x - 8$$
3. Group like terms:
$$= (6x^3 - 2x^3) + (-4x^2 + 9x^2) + (7x + 3x) + (-2 - 8)$$
4. Perform arithmetic:
$$= 4x^3 + 5x^2 + 10x - 10$$
Answer: $$4x^3 + 5x^2 + 10x - 10$$
Question 4 (Part ii): Subtract polynomial $Q(y) = 4y^4 + 3y^3 - 6y^2 + 10$ from $P(y) = 9y^4 - 2y^2 + 5y - 12$.
Solution:
1. Formulate subtraction $P(y) - Q(y)$:
$$P(y) - Q(y) = (9y^4 - 2y^2 + 5y - 12) - (4y^4 + 3y^3 - 6y^2 + 10)$$
2. Change signs of each term of $Q(y)$:
$$= 9y^4 - 2y^2 + 5y - 12 - 4y^4 - 3y^3 + 6y^2 - 10$$
3. Align like terms:
$$= (9y^4 - 4y^4) + (-3y^3) + (-2y^2 + 6y^2) + 5y + (-12 - 10)$$
4. Simplify:
$$= 5y^4 - 3y^3 + 4y^2 + 5y - 22$$
Answer: $$5y^4 - 3y^3 + 4y^2 + 5y - 22$$
Question 5 (Part i): Find the product of polynomials $P(x) = 2x - 3$ and $Q(x) = 3x^2 + 4x - 5$.
Solution:
1. Apply distributive property:
$$P(x) \cdot Q(x) = (2x - 3)(3x^2 + 4x - 5)$$
$$= 2x(3x^2 + 4x - 5) - 3(3x^2 + 4x - 5)$$
2. Expand each term:
$$= (2x \cdot 3x^2 + 2x \cdot 4x + 2x \cdot (-5)) + (-3 \cdot 3x^2 - 3 \cdot 4x - 3 \cdot (-5))$$
$$= (6x^3 + 8x^2 - 10x) - (9x^2 + 12x - 15)$$
$$= 6x^3 + 8x^2 - 10x - 9x^2 - 12x + 15$$
3. Combine like terms:
$$= 6x^3 + (8x^2 - 9x^2) + (-10x - 12x) + 15$$
$$= 6x^3 - x^2 - 22x + 15$$
Answer: $$6x^3 - x^2 - 22x + 15$$
Question 5 (Part ii): Multiply $P(x, y) = x^2 - xy + y^2$ by $Q(x, y) = x + y$.
Solution:
1. Set up multiplication:
$$(x + y)(x^2 - xy + y^2)$$
2. Distribute $x$ and $y$ separately across the trinomial:
$$= x(x^2 - xy + y^2) + y(x^2 - xy + y^2)$$
$$= (x^3 - x^2y + xy^2) + (x^2y - xy^2 + y^3)$$
3. Group like terms:
$$= x^3 + (-x^2y + x^2y) + (xy^2 - xy^2) + y^3$$
$$= x^3 + 0 + 0 + y^3 = x^3 + y^3$$
Answer: $$x^3 + y^3$$
Question 6 (Part i): Evaluate the polynomial $P(x) = 2x^3 - 5x^2 + 3x - 8$ for $x = -2$.
Solution:
1. Substitute $x = -2$ into $P(x)$:
$$P(-2) = 2(-2)^3 - 5(-2)^2 + 3(-2) - 8$$
2. Calculate powers of $-2$:
$$(-2)^3 = -8, \quad (-2)^2 = 4$$
3. Substitute back and multiply:
$$P(-2) = 2(-8) - 5(4) + 3(-2) - 8$$
$$= -16 - 20 - 6 - 8$$
4. Perform addition of negative integers:
$$= -50$$
Answer: $$-50$$
Question 6 (Part ii): Evaluate the polynomial $P(x, y) = 3x^2y - 2xy^2 + 4x - 5y + 7$ at $x = 3$ and $y = -1$.
Solution:
1. Substitute $x = 3$ and $y = -1$ into the polynomial:
$$P(3, -1) = 3(3)^2(-1) - 2(3)(-1)^2 + 4(3) - 5(-1) + 7$$
2. Simplify exponent terms:
$$(3)^2 = 9, \quad (-1)^2 = 1$$
3. Perform multiplications:
$$= 3(9)(-1) - 2(3)(1) + 12 + 5 + 7$$
$$= -27 - 6 + 12 + 5 + 7$$
4. Combine numerical results:
$$= (-27 - 6) + (12 + 5 + 7)$$
$$= -33 + 24 = -9$$
Answer: $$-9$$