Fundamental Concepts of Algebra
Question 1. Separate the constants and variables from the following list:
${(-5), (10 + x), (y + 7), (9x + yz), (5.8 y - 3x), (8 - 4), (7x - 6y \times p),
(3x^2 - 5y), (\sqrt{3}), (\dfrac{3yz}{4}), (\sqrt{xy}), 111}$
Constants are fixed values, following are the constants:
$\Rightarrow (-5), (\sqrt{3}), (8 - 4), 111$
Variables are values that do not contain fixed values. Their values can change overtime:
$\Rightarrow (10 + x), (y + 7), (9x + yz), (5.8 y - 3x), (7x - 6y \times p), (\dfrac{3yz}{4}), (\sqrt{xy}), (3x^2 - 5y)$
Question 2. Write down the number of terms in given polynomials:
(i) $ 7x^2 + 3 \times xy$
$\Rightarrow$ Contains 2 terms that include: $7x^2$ and $3xy$
(ii) $yz \div 3 - 7$
$\Rightarrow$ Contains 2 terms that include: $\dfrac{yz}{3}$ and $7$
(iii) $3x - ay + y \times z$
$\Rightarrow$ Contains 3 terms that include: $3x$, $ay$ and $yz$
(iv) $54 + x \times y \div 4$
$\Rightarrow$ Contains 2 terms that include: 54 and $\dfrac{xy}{4}$
(v) $7x - 6y \times p$
$\Rightarrow$ Contains 2 terms that include: $7x$ and $6py$
Question 3. Separate monomials, binomials, trinomials and polynomials from the following:
(i) $5x - 2x^2 + 17x^3 -9$
$\Rightarrow$ Polynomial containing more than 2 terms that include: $5x$, $2x^2$, $17x^3$ and 9
(ii) $2y^2 - 5y + 7$
$\Rightarrow$ Trinomial containing 3 terms that include: $2y^2$, $5y$ and 7
(iii) $4x \times 6y$
$\Rightarrow$ Monomial containing a single term that include: $24xy$
(iv) $3x^2 + 8yz$
$\Rightarrow$ Binomial containing 2 terms that include: $3x^2$ and $8yz$
Question 4. Mention the degree of each polynomial given below:
(i) $a + ab + b^2$
$\Rightarrow$ The degree of $a$ equals 1, the degree of $ab$ equals $(1 + 1 =2)$ and degree of $b^2$ equals 2. Hence the highest degree is 2.
(ii) $x^2 - y^3 + 5$
$\Rightarrow$ The degree of $x^2$ equals 2 and the degree of $y$ equals $3$. Hence the highest degree is 3.
(iii) $abc - 6$
$\Rightarrow$ The degree of $abc$ equals $(1 + 1 + 1 = 3)$. Hence the highest degree is 3.
(iv) $ab - 3$
$\Rightarrow$ The degree of $ab$ equals $(1 + 1 = 2)$. Hence the highest degree is 2.
(v) $y^2 - 6xy^3 + 5x^2y^8$
$\Rightarrow$ The degree of $y^2$ equals 2, the degree of $6xy^3$ equals $(1 + 3 = 4)$ and degree of $5x^2y^8$ equals $(2 + 8 = 10)$. Hence the highest degree is 10.
(vi) $a^3b + bc^2 - zx^4$
$\Rightarrow$ The degree of $a^2b$ equals $(2 + 1 = 3)$, the degree of $bc^2$ equals $(1 + 2 = 3)$ and degree of $zx^4$ equals $(1 + 4 = 5)$. Hence the highest degree is 5.
(vii) $x^3y^4 - 6x^2y^3 + 9x^4y^2z^5$
$\Rightarrow$ The degree of $x^3y^4$ equals $(3 + 4 = 7)$, the degree of $6x^2y^3$ equals $(2 + 3 = 5)$ and degree of $9x^4y^2z^5$ equals $(4 + 2 + 5 = 11)$. Hence the highest degree is 11.
Question 5. Mention the coefficient of the following:
(i) $ab$ in $7abx$
$\Rightarrow$ In $7abx = 7x$ is the coefficient of $ab$.
(ii) $7a$ in $7abx$
$\Rightarrow$ In $7abx = 7a$ is the coefficient of $bx$.
(iii) $2x^3$ in $2x^3 - 2x$
$\Rightarrow$ In $2x^3 - 2x =$ 1 is the coefficient of $2x^2$.
(iv) $4$ in $a^2 - 4ax + a$
$\Rightarrow$ In $a^2 - 4ax + a = (-ax)$ is the coefficient of 4.
(v) $7xy$ in $x^2 - 7xy + y^2$
$\Rightarrow$ In $x^2 - 7xy + y^2 = (-1)$ is the coefficient of $7xy$.
Question 6. In $\dfrac{3}{7}x^2yz^3$, write the coefficient of:
(i) 3
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{1}{7}x^2yz^3$ is the coefficient of 3.
(ii) $\dfrac{3}{7}$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = x^2yz^3$ is the coefficient of $\dfrac{3}{7}$.
(iii) $3x^2$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{1}{7}yz^3$ is the coefficient of $3x^2$.
(iv) $x^2y$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{3}{7}z^3$ is the coefficient of $x^2y$.
(v) $z^3$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{3}{7}x^2y$ is the coefficient of $z^3$.
(vi) $x^2z^3$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{3}{7}y$ is the coefficient of $x^2z^3$.
(vii) $\dfrac{1}{7}yz^3$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = 3x^2$ is the coefficient of $\dfrac{1}{7}yz^3$.
(viii) $xz$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{3}{7}xyz^2$ is the coefficient of $xz$.
(ix) $3xyz^2$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = \dfrac{1}{7}xyz$ is the coefficient of $3xyz^2$.
(ix) $\dfrac{1}{7}xyz$
$\Rightarrow$ In $\dfrac{3}{7}x^2yz^3 = 3xz^2$ is the coefficient of $\dfrac{1}{7}xyz$.
Question 7. From the list below separate the Like Terms:
(i) ${8xy, (-3yx^2), xy^2, 3.2x^2y, (-7yx), 5.4y^2x, x^2y}$
$\Rightarrow$ Like terms are:
(a) $8xy$ and $(-7yx)$
(b) $3.2x^2y, (-3yx^2)$ and $(x^2y)$
(c) $5.4y^2x$ and $xy^2$
(ii) ${y^2z^3, xy^2z^3, 2x^2yz, (-3y^2z^3), (-7xz^3y^2), 4x^2yz, 8z^3y^2}$
$\Rightarrow$ Like terms are:
(a) $y^2z^3, (-3y^2z^3)$ and $8z^3y^2$
(b) $(-7xz^3y^2)$ and $(-3y^2xz^3)$
(c) $2x^2yz$ and $4x^2yz$