Fundamental Concepts of Algebra — Exercise 12(A)

Separate the constants and the variables from each of the following:

6, $4y, −3x, \dfrac{5}{4}, \dfrac{4}{5}xy, az, 7p$, 0, $\dfrac{9x}{y}, \dfrac{3}{4x}, -\dfrac{xz}{3y}$

Answer

A symbol having a fixed numerical value in all situations is a constant, whereas a symbol whose value changes with the situation is a variable. Also, every combination of a constant and a variable is a variable.

Constants = 6, $\mathbf{\dfrac{5}{4}}$ and 0

Variables = $\bm{4y, -3x, \dfrac{4}{5}xy, az, 7p, \dfrac{9x}{y}, \dfrac{3}{4x}}$ and $\bm{-\dfrac{xz}{3y}}$

Group the like terms together:

$4x, -3y, -x, \dfrac{2}{3}x, \dfrac{4}{5}y$ and $y$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{4x, -x}$ and $\bm{\dfrac{2}{3}x}; \bm{-3y, \dfrac{4}{5}y}$ and $\bm{y}$.

Group the like terms together:

$\dfrac{2}{3}xy, -4yx, 2yz, \dfrac{-2}{3}yz, \dfrac{zy}{3}$ and $yx$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{\dfrac{2}{3}xy, -4yx}$ and $\bm{yx}; \bm{2yz, \dfrac{-2}{3}yz}$ and $\bm{\dfrac{zy}{3}}$.

Group the like terms together:

$-ab^2, b^2a^2, 7b^2a, -3a^2b^2$ and $2ab^2$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{-ab^2, 7b^2a}$ and $\bm{2ab^2}; \bm{b^2a^2}$ and $\bm{-3a^2b^2}$.

Group the like terms together:

$5ax, -5by, \dfrac{by}{7}, 7xa$ and $\dfrac{2ax}{3}$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{5ax, 7xa}$ and $\bm{\dfrac{2ax}{3}}; \bm{-5by}$ and $\bm{\dfrac{by}{7}}$.

State whether true or false:

(i) 16 is a constant and $y$ is a variable, but $16y$ is variable.

(ii) $5x$ has two terms 5 and $x$.

(iii) The expression 5 + $x$ has two terms 5 and $x$.

(iv) The expression $2x^2 + x$ is a trinomial.

(v) $ax^2 + bx + c$ is a trinomial.

(vi) $8 \times ab$ is a binomial.

(vii) 8 + $ab$ is a binomial.

(viii) $x^3 − 5xy + 6x + 7$ is a polynomial.

(ix) $x^3 − 5xy + 6x + 7$ is a multinomial.

(x) The coefficient of $x$ in $5x$ is $5x$.

(xi) The coefficient of $ab$ in $−ab$ is −1.

(xii) The coefficient of $y$ in $−3xy$ is −3.

Answer

(i) True, as the value of $16y$ changes with the value of $y$, so every combination of a constant and a variable is a variable.

(ii) False, as $5x$ is a single term. The sign of multiplication does not separate the terms.

(iii) True, as terms are separated by the signs + and - only, so $5 + x$ has two terms 5 and $x$.

(iv) False, as $2x^2 + x$ has two unlike terms, so it is a binomial.

(v) True, as $ax^2 + bx + c$ has three unlike terms.

(vi) False, as $8 \times ab = 8ab$ has only one term, so it is a monomial.

(vii) True, as $8 + ab$ has two unlike terms.

(viii) True, as in $x^3 - 5xy + 6x + 7$ the sum of the powers of the variables in each term is a whole number.

(ix) True, as $x^3 - 5xy + 6x + 7$ has more than two terms.

(x) False, as $5x = 5 \times x$, so the coefficient of $x$ in $5x$ is 5.

(xi) True, as $-ab = (-1) \times ab$, so the coefficient of $ab$ in $-ab$ is -1.

(xii) False, as $-3xy = (-3x) \times y$, so the coefficient of $y$ in $-3xy$ is $-3x$.

State the number of terms in the following expression:

$2a − b$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$2a - b$ has the terms $2a$ and $-b$ → 2 terms.

State the number of terms in the following expression:

$3 \times x + \dfrac{a}{2}$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$3 \times x + \dfrac{a}{2} = 3x + \dfrac{a}{2}$, which has the terms $3x$ and $\dfrac{a}{2}$ → 2 terms.

State the number of terms in the following expression:

$3x − \dfrac{x}{p}$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$3x - \dfrac{x}{p}$ has the terms $3x$ and $-\dfrac{x}{p}$ → 2 terms.

State the number of terms in the following expression:

$a \div x \times b + c$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$a \div x \times b + c = \dfrac{ab}{x} + c$, which has the terms $\dfrac{ab}{x}$ and $c$ → 2 terms.

State the number of terms in the following expression:

$3x \div 2 + y + 4$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$3x \div 2 + y + 4 = \dfrac{3x}{2} + y + 4$, which has the terms $\dfrac{3x}{2}, y$ and 4 → 3 terms.

State the number of terms in the following expression:

$xy \div 2$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$xy \div 2 = \dfrac{xy}{2}$, which has only one term → 1 term.

State the number of terms in the following expression:

$x + y \div a$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$x + y \div a = x + \dfrac{y}{a}$, which has the terms $x$ and $\dfrac{y}{a}$ → 2 terms.

State the number of terms in the following expression:

$2x + y + 8 \div y$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$2x + y + 8 \div y = 2x + y + \dfrac{8}{y}$, which has the terms $2x, y$ and $\dfrac{8}{y}$ → 3 terms.

State the number of terms in the following expression:

$2 \times a + 3 \div b + 4$

Answer

Terms are separated by the signs +(plus) and -(minus) only. The signs ×(multiplication) and ÷(division) do not separate the terms.

$2 \times a + 3 \div b + 4 = 2a + \dfrac{3}{b} + 4$, which has the terms $2a, \dfrac{3}{b}$ and 4 → 3 terms.

State whether true or false:

(i) $xy$ and $−yx$ are like terms.

(ii) $x^2y$ and $−y^2x$ are like terms.

(iii) $a$ and $−a$ are like terms.

(iv) $−ba$ and $2ab$ are unlike terms.

(v) 5 and $5x$ are like terms.

(vi) $3xy$ and $4xyz$ are unlike terms.

Answer

The terms having the same literal coefficients are like terms, and the terms that do not have the same literal coefficients are unlike terms.

(i) True, as the literal coefficient of each of $xy$ and $-yx$ is $xy$.

(ii) False, as the literal coefficients $x^2y$ and $y^2x$ are different, so they are unlike terms.

(iii) True, as the literal coefficient of each of $a$ and $-a$ is $a$.

(iv) False, as the literal coefficient of each of $-ba$ and $2ab$ is $ab$, so they are like terms.

(v) False, as 5 has no literal coefficient while the literal coefficient of $5x$ is $x$, so they are unlike terms.

(vi) True, as the literal coefficients $xy$ and $xyz$ are different.

For each expression given below, state whether it is a monomial, or a binomial or a trinomial.

(i) $xy$

(ii) $xy + x$

(iii) $2x \div y$

(iv) $−a$

(v) $ax^2 − x + 5$

(vi) $−3bc + d$

(vii) $1 + x + y$

(viii) $1 + x \div y$

(ix) $x + xy − y^2$

Answer

An expression with only one term is a monomial, with two unlike terms is a binomial and with three unlike terms is a trinomial.

(i) $xy$ has 1 term → Monomial.

(ii) $xy + x$ has 2 terms → Binomial.

(iii) $2x \div y = \dfrac{2x}{y}$, which has 1 term → Monomial.

(iv) $-a$ has 1 term → Monomial.

(v) $ax^2 - x + 5$ has 3 terms → Trinomial.

(vi) $-3bc + d$ has 2 terms → Binomial.

(vii) $1 + x + y$ has 3 terms → Trinomial.

(viii) $1 + x \div y = 1 + \dfrac{x}{y}$, which has 2 terms → Binomial.

(ix) $x + xy - y^2$ has 3 terms → Trinomial.

Write down the coefficient of $x$ in the following monomials:

(i) $x$

(ii) $−x$

(iii) $−3x$

(iv) $−5ax$

(v) $\dfrac{3}{2}xy$

(vi) $\dfrac{ax}{y}$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

(i) $x = 1 \times x$, so the coefficient of $x$ is 1.

(ii) $-x = (-1) \times x$, so the coefficient of $x$ is −1.

(iii) $-3x = (-3) \times x$, so the coefficient of $x$ is −3.

(iv) $-5ax = (-5a) \times x$, so the coefficient of $x$ is $\bm{-5a}$.

(v) $\dfrac{3}{2}xy = \left(\dfrac{3}{2}y\right) \times x$, so the coefficient of $x$ is $\bm{\dfrac{3}{2}y}$.

(vi) $\dfrac{ax}{y} = \left(\dfrac{a}{y}\right) \times x$, so the coefficient of $x$ is $\bm{\dfrac{a}{y}}$.

Write the coefficient of:

$x$ in $−3xy^2$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-3xy^2 = (-3y^2) \times x$, so the coefficient of $x$ is $\bm{-3y^2}$.

Write the coefficient of:

$x$ in $−ax$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-ax = (-a) \times x$, so the coefficient of $x$ is $\bm{-a}$.

Write the coefficient of:

$y$ in $−y$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-y = (-1) \times y$, so the coefficient of $y$ is −1.

Write the coefficient of:

$y$ in $\dfrac{2}{a}y$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$\dfrac{2}{a}y = \left(\dfrac{2}{a}\right) \times y$, so the coefficient of $y$ is $\bm{\dfrac{2}{a}}$.

Write the coefficient of:

$xy$ in $−2xyz$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-2xyz = (-2z) \times xy$, so the coefficient of $xy$ is $\bm{-2z}$.

Write the coefficient of:

$ax$ in $−axy^2$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-axy^2 = (-y^2) \times ax$, so the coefficient of $ax$ is $\bm{-y^2}$.

Write the coefficient of:

$x^2y$ in $−3ax^2y$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-3ax^2y = (-3a) \times x^2y$, so the coefficient of $x^2y$ is $\bm{-3a}$.

Write the coefficient of:

$xy^2$ in $5axy^2$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$5axy^2 = (5a) \times xy^2$, so the coefficient of $xy^2$ is $\bm{5a}$.

State the numeral coefficient of the following monomial:

$5xy$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

The numeral coefficient of $5xy$ is 5.

State the numeral coefficient of the following monomial:

$abc$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

$abc = 1 \times abc$, so the numeral coefficient of $abc$ is 1.

State the numeral coefficient of the following monomial:

$5pqr$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

The numeral coefficient of $5pqr$ is 5.

State the numeral coefficient of the following monomial:

$\dfrac{-2x}{y}$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

$\dfrac{-2x}{y} = (-2) \times \dfrac{x}{y}$, so the numeral coefficient is −2.

State the numeral coefficient of the following monomial:

$\dfrac{2}{3}xy^2$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

The numeral coefficient of $\dfrac{2}{3}xy^2$ is $\mathbf{\dfrac{2}{3}}$.

State the numeral coefficient of the following monomial:

$\dfrac{-15xy}{2z}$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

$\dfrac{-15xy}{2z} = \left(\dfrac{-15}{2}\right) \times \dfrac{xy}{z}$, so the numeral coefficient is $\mathbf{\dfrac{-15}{2}}$.

State the numeral coefficient of the following monomial:

$−7x \div y$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

$-7x \div y = \dfrac{-7x}{y} = (-7) \times \dfrac{x}{y}$, so the numeral coefficient is −7.

State the numeral coefficient of the following monomial:

$−3x \div (2y)$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

$-3x \div (2y) = \dfrac{-3x}{2y} = \left(\dfrac{-3}{2}\right) \times \dfrac{x}{y}$, so the numeral coefficient is $\mathbf{\dfrac{-3}{2}}$.

Write the degree of each of the following polynomials:

$x + x^2$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms.

Degrees of the terms are 1 and 2. The greatest is 2.

Hence, the degree of the polynomial = 2.

Write the degree of each of the following polynomials:

$5x^2 - 7x + 2$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms.

Degrees of the terms are 2, 1 and 0. The greatest is 2.

Hence, the degree of the polynomial = 2.

Write the degree of each of the following polynomials:

$x^3 - x^8 + x^{10}$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms.

Degrees of the terms are 3, 8 and 10. The greatest is 10.

Hence, the degree of the polynomial = 10.

Write the degree of each of the following polynomials:

$1 - 100x^{20}$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms.

Degrees of the terms are 0 and 20. The greatest is 20.

Hence, the degree of the polynomial = 20.

Write the degree of each of the following polynomials:

$4 + 4x - 4x^3$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms.

Degrees of the terms are 0, 1 and 3. The greatest is 3.

Hence, the degree of the polynomial = 3.

Write the degree of each of the following polynomials:

$8x^2y - 3y^2 + x^2y^5$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms. For a term containing more than one variable, the power of that term is the sum of the powers of all the variables in it.

Degrees of the terms are (2 + 1) = 3, 2 and (2 + 5) = 7. The greatest is 7.

Hence, the degree of the polynomial = 7.

Write the degree of each of the following polynomials:

$8z^3 - 8y^2z^3 + 7yz^5$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms. For a term containing more than one variable, the power of that term is the sum of the powers of all the variables in it.

Degrees of the terms are 3, (2 + 3) = 5 and (1 + 5) = 6. The greatest is 6.

Hence, the degree of the polynomial = 6.

Write the degree of each of the following polynomials:

$4y^2 - 3x^3 + y^2x^7$

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms. For a term containing more than one variable, the power of that term is the sum of the powers of all the variables in it.

Degrees of the terms are 2, 3 and (2 + 7) = 9. The greatest is 9.

Hence, the degree of the polynomial = 9.