Algebraic Expressions - Exercise 13(B)

Add the following expressions:

(i) 2x², -5x², -x², 6x²

(ii) x² - 2xy + 3y², 5y² + 3xy - 6x²

(iii) 2x + 9y - 7z, 3y + z - 3x, 2z - 4y - x

(iv) 2ab + 3bc - 5ca, 4bc - 3ab + 7ca, 2ca - ab - 5bc

(v) 3x³ + 2x² - 6x + 3, 2x³ - 3x² - x - 4, 1 + 2x - 3x² - 4x³

(vi) 3n² + 5mn - 6m², 2m² - 3mn - 4n², 2mn - 3m² - 7n²

(vii) 3z³ - z² + 5, 1 - 2z + z², 3 + 2z - z³

Answer

(i) 2x², -5x², -x², 6x²

Since these are all like terms, we can stack them in a single column:

$\begin{array}{r} 2x^2 \\ -5x^2 \\ -\phantom{5} x^2 \\ +6x^2 \\ \hline 2x^2 \\ \hline \end{array}$

Hence, the answer is 2x²

(ii) x² - 2xy + 3y², 5y² + 3xy - 6x²

Arranging the expressions so that x² is under x², xy is under xy, and y² is under y²:

$\begin{array}{rcccc} x^2 & - & 2xy & + & 3y^2 \\ -6x^2 & + & 3xy & + & 5y^2 \\ \hline -5x^2 & + & xy & + & 8y^2 \\ \hline \end{array}$

Hence, the answer is -5x² + xy + 8y²

(iii) 2x + 9y - 7z, 3y + z - 3x, 2z - 4y - x

Arranging the expressions so that x is under x, y is under y, and z is under z:

$\begin{array}{rcccc} 2x & + & 9y & - & 7z \\ -3x & + & 3y & + & z \\ -\phantom{3} x & - & 4y & + & 2z \\ \hline -2x & + & 8y & - & 4z \\ \hline \end{array}$

Hence, the answer is -2x + 8y - 4z

(iv) 2ab + 3bc - 5ca, 4bc - 3ab + 7ca, 2ca - ab - 5bc

Arranging the expressions so that ab is under ab, bc is under bc, and ca is under ca:

$\begin{array}{rcccc} 2ab & + & 3bc & - & 5ca \\ -3ab & + & 4bc & + & 7ca \\ -\phantom{3} ab & - & 5bc & + & 2ca \\ \hline -2ab & + & 2bc & + & 4ca \\ \hline \end{array}$

Hence, the answer is -2ab + 2bc + 4ca

(v) 3x³ + 2x² - 6x + 3, 2x³ - 3x² - x - 4, 1 + 2x - 3x² - 4x³

Arranging the expressions into descending powers of x (x³, x², x, constant):

$\begin{array}{rcccccc} 3x^3 & + & 2x^2 & - & 6x & + & 3 \\ +2x^3 & - & 3x^2 & - & x & - & 4 \\ -4x^3 & - & 3x^2 & + & 2x & + & 1 \\ \hline x^3 & - & 4x^2 & - & 5x & + & 0 \\ \hline \end{array}$

Hence, the answer is x³ - 4x² - 5x

(vi) 3n² + 5mn - 6m², 2m² - 3mn - 4n², 2mn - 3m² - 7n²

Arranging the expressions so that m² is under m², mn is under mn, and n² is under n²:

$\begin{array}{rcccc} -6m^2 & + & 5mn & + & 3n^2 \\ +2m^2 & - & 3mn & - & 4n^2 \\ -3m^2 & + & 2mn & - & 7n^2 \\ \hline -7m^2 & + & 4mn & - & 8n^2 \\ \hline \end{array}$

Hence, the answer is -7m² + 4mn - 8n²

(vii) 3z³ - z² + 5, 1 - 2z + z², 3 + 2z - z³

Arranging the expressions in descending powers of z and use 0 as a placeholder for any missing terms:

$\begin{array}{rcccccc} 3z^3 & - & z^2 & + & 0 & + & 5 \\ +\phantom{3} 0 & + & z^2 & - & 2z & + & 1 \\ -z^3 & + & 0 & + & 2z & + & 3 \\ \hline 2z^3 & + & 0 & + & 0 & + & 9 \\ \hline \end{array}$

Hence, the answer is 2z³ + 9

Simplify:

(i) 5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x

(ii) 4x³ - 2x² + 5x - 1 + 8x + x² - 6x³ + 7 - 6x + 3 - 3x² - x³

(iii) 2x² + 3xy - 3y² + x² - xy + y²

(iv) 2 - 3z² + 5yz + 7y² - 8 + z² - 6yz - 9y² + 1 - 2z² - 2yz - y²

(v) 2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p

Answer

(i) 5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x

Arranging the like terms together, we have:

5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x

= (5x - 3x - 2x) + (3y + 2y - 7y) + (-8z + 5z + z)

Combining coefficients:

= (5 - 3 - 2)x + (3 + 2 - 7)y + (-8 + 5 + 1)z

= (0)x + (-2)y + (-2)z

= -2y - 2z

Hence, the answer is -2y - 2z

(ii) 4x³ - 2x² + 5x - 1 + 8x + x² - 6x³ + 7 - 6x + 3 - 3x² - x³

Arranging the like terms together, we have:

4x³ - 2x² + 5x - 1 + 8x + x² - 6x³ + 7 - 6x + 3 - 3x² - x³

= (4x³ - 6x³ - x³) + (-2x² + x² - 3x²) + (5x + 8x - 6x) + (-1 + 7 + 3)

Combining coefficients:

= (4 - 6 - 1)x³ + (-2 + 1 - 3)x² + (5 + 8 - 6)x + (9)

= (-3)x³ + (-4)x² + (7)x + 9

= -3x³ - 4x² + 7x + 9

Hence, the answer is -3x³ - 4x² + 7x + 9

(iii) 2x² + 3xy - 3y² + x² - xy + y²

Arranging the like terms together, we have:

2x² + 3xy - 3y² + x² - xy + y²

= (2x² + x²) + (3xy - xy) + (-3y² + y²)

Combining coefficients:

= (2 + 1)x² + (3 - 1)xy + (-3 + 1)y²

= 3x² + 2xy - 2y²

Hence, the answer is 3x² + 2xy - 2y²

(iv) 2 - 3z² + 5yz + 7y² - 8 + z² - 6yz - 9y² + 1 - 2z² - 2yz - y²

Arranging the like terms together, we have:

2 - 3z² + 5yz + 7y² - 8 + z² - 6yz - 9y² + 1 - 2z² - 2yz - y²

= (2 - 8 + 1) + (-3z² + z² - 2z²) + (5yz - 6yz - 2yz) + (7y² - 9y² - y²)

Combining coefficients:

= (-5) + (-3 + 1 - 2)z² + (5 - 6 - 2)yz + (7 - 9 - 1)y²

= -5 + (-4)z² + (-3)yz + (-3)y²

= -5 - 4z² - 3yz - 3y²

Hence, the answer is -5 - 4z² - 3yz - 3y²

(v) 2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p

Arranging the like terms together, we have:

2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p

= (2m + 2m - 3m) + (-3n + n - 4n) + (5p - 2p + p)

Combining coefficients:

= (2 + 2 - 3)m + (-3 + 1 - 4)n + (5 - 2 + 1)p

= (1)m + (-6)n + (4)p

= m - 6n + 4p

Hence, the answer is m - 6n + 4p

The two adjacent sides of a rectangle are 3a - b and 6b - a. Find its perimeter.

Answer

Given:

Length = (3a - b)

Breadth = (6b - a)

Perimeter of a rectangle = 2 x (Length + Breadth)

Substituting the values above, we get:

Perimeter of a rectangle = 2 x (3a - b + 6b - a)

= 2 x [(3a - a) + (-b + 6b)] $\quad$[Grouping like terms]

= 2 x (2a + 5b) $\quad$[Distributive property of multiplication]

= 4a + 10b

The perimeter of the rectangle is 4a + 10b.

Find the perimeter of a triangle whose sides are 2y + 3z, z - y, 4y - 2z.

Answer

Given:

Side 1 = 2y + 3z

Side 2 = z - y

Side 3 = 4y - 2z

Perimeter of a triangle = Side 1 + Side 2 + Side 3

Substituting the values above, we get:

Perimeter of a triangle = (2y + 3z + z - y + 4y - 2z)

= (2y - y + 4y) + (3z + z - 2z) $\quad$[Grouping like terms]

= (2 - 1 + 4)y + (3 + 1 - 2)z $\quad$[Combining coefficients]

= 5y + 2z

The perimeter of the triangle is 5y + 2z.

Subtract:

(i) 3a - 2b + 4c from 5a - 3b - 5c

(ii) 5x² - 3xy - 7y² from 3x² - xy - 2y²

(iii) 3p³ - 5p²q + 2q² from q² + p²q - 4p³

(iv) ab - bc - ca from 3ab + 2bc - 4ca

(v) 3z³ - 2z² + 7z - 8 from 8 - z - z²

(vi) 2abc - a² - b² from b² + a² - 2abc

Answer

(i) 3a - 2b + 4c from 5a - 3b - 5c

We have:

$\begin{array}{rcccc} 5a & - & 3b & - & 5c \\ +3a & - & 2b & + & 4c \\ -\phantom{3a} & + & & - \\ \hline 2a & - & b & - & 9c \\ \hline \end{array}$

Hence, the answer is 2a - b - 9c

(ii) 5x² - 3xy - 7y² from 3x² - xy - 2y²

We have:

$\begin{array}{rcccc} 3x^2 & - & xy & - & 2y^2 \\ +5x^2 & - & 3xy & - & 7y^2 \\ -\phantom{5x^2} & + & & + \\ \hline -2x^2 & + & 2xy & + & 5y^2 \\ \hline \end{array}$

Hence, the answer is -2x² + 2xy + 5y²

(iii) 3p³ - 5p²q + 2q² from q² + p²q - 4p³

Arranging the terms to match (p³, p²q, q²):

$\begin{array}{rcc} -4p^3 & + & p^2q & + & q^2 \\ +3p^3 & - & 5p^2q & + & 2q^2 \\ -\phantom{3p^3} & + & & - \\ \hline -7p^3 & + & 6p^2q & - & q^2 \\ \hline \end{array}$

Hence, the answer is -7p³ + 6p²q - q²

(iv) ab - bc - ca from 3ab + 2bc - 4ca

Arranging the terms to match(ab, bc, ca):

$\begin{array}{rcccc} 3ab & + & 2bc & - & 4ca \\ +ab & - & bc & - & ca \\ -\phantom{ab} & + & & + \\ \hline 2ab & + & 3bc & - & 3ca \\ \hline \end{array}$

Hence, the answer is 2ab + 3bc - 3ca

(v) 3z³ - 2z² + 7z - 8 from 8 - z - z²

Arranging the expressions in ascending powers of z and use 0 as a placeholder for any missing terms:

$\begin{array}{rcccccc} 8 & - & z & - & z^2 & + & 0 \\ -8 & + & 7z & - & 2z^2 & + & 3z^3 \\ +\phantom{8} & - & & + & & - \\ \hline 16 & - & 8z & + & z^2 & - & 3z^3 \\ \hline \end{array}$

Hence, the answer is 16 - 8z + z² - 3z³

(vi) 2abc - a² - b² from b² + a² - 2abc

Arranging the terms to match (a², b², abc):

$\begin{array}{rcc} a^2 & + & b^2 & - & 2abc \\ -a^2 & - & b^2 & + & 2abc \\ +\phantom{a^2} & + & & - \\ \hline 2a^2 & + & 2b^2 & - & 4abc \\ \hline \end{array}$

Hence, the answer is 2a² + 2b² - 4abc

(i) Subtract 6x³ - 5x² + 4x - 3 from the sum of x + 2x² - 3x³ and 2 - x² + 6x - x³.

(ii) Subtract the sum of a + 2b - 3c and 2c - 3b - 4a from the sum of 5b - 4c + a and 2c - 3b - 4a.

(iii) Subtract the sum of x² - 5xy + 2y² and y² - 2xy - 3x² from the sum of 6x² - 8xy - y² and 2xy - 2y² - x².

Answer

(i) Subtract 6x³ - 5x² + 4x - 3 from the sum of x + 2x² - 3x³ and 2 - x² + 6x - x³.

Let's find the sum of the last two expressions:

We have:

x + 2x² - 3x³ and 2 - x² + 6x - x³

Arranging the terms in descending powers of x and use 0 as a placeholder for any missing term:

$\begin{array}{rcccccc} -3x^3 & + & 2x^2 & + & x & + & 0 \\ -\phantom{3} x^3 & - & x^2 & + & 6x & + & 2 \\ \hline -4x^3 & + & x^2 & + & 7x & + & 2 \\ \hline \end{array}$

The sum is -4x³ + x² + 7x + 2

Now, subtract 6x³ - 5x² + 4x - 3 from the above sum:

$\begin{array}{rcccccc} -4x^3 & + & x^2 & + & 7x & + & 2 \\ +6x^3 & - & 5x^2 & + & 4x & - & 3 \\ -\phantom{6x^3} & + & & - & & + \\ \hline -10x^3 & + & 6x^2 & + & 3x & + & 5 \\ \hline \end{array}$

Hence, the answer is -10x³ + 6x² + 3x + 5

(ii) Subtract the sum of a + 2b - 3c and 2c - 3b - 4a from the sum of 5b - 4c + a and 2c - 3b - 4a.

Let's find the sum of first pair:

We have:

a + 2b - 3c and 2c - 3b - 4a

Arranging the terms to match (a, b, c):

$\begin{array}{rcccc} a & + & 2b & - & 3c \\ -4a & - & 3b & + & 2c \\ \hline -3a & - & b & - & c \\ \hline \end{array}$

Sum 1 = -3a - b - c

Now, let's find the sum of second pair:

We have:

5b - 4c + a and 2c - 3b - 4a

Arranging the terms to match (a, b, c):

$\begin{array}{rcccc} a & + & 5b & - & 4c \\ -4a & - & 3b & + & 2c \\ \hline -3a & + & 2b & - & 2c \\ \hline \end{array}$

Sum 2 = -3a + 2b - 2c

Let's subtract sum 1 from sum 2:

$\begin{array}{rcccc} -3a & + & 2b & - & 2c \\ -3a & - & b & - & c \\ +\phantom{3a} & + & & + \\ \hline 0 & + & 3b & - & c \\ \hline \end{array}$

Hence, the answer is 3b - c

(iii) Subtract the sum of x² - 5xy + 2y² and y² - 2xy - 3x² from the sum of 6x² - 8xy - y² and 2xy - 2y² - x².

Let's find the sum of first pair:

We have:

x² - 5xy + 2y² and y² - 2xy - 3x²

Arranging the terms to match (x², xy, y²):

$\begin{array}{rcccc} x^2 & - & 5xy & + & 2y^2 \\ -3x^2 & - & 2xy & + & y^2 \\ \hline -2x^2 & - & 7xy & + & 3y^2 \\ \hline \end{array}$

Sum 1 = -2x² - 7xy + 3y²

Now, let's find the sum of second pair:

We have:

6x² - 8xy - y² and 2xy - 2y² - x²

Arranging the terms to match (x², xy, y²):

$\begin{array}{rcccc} 6x^2 & - & 8xy & - & y^2 \\ -x^2 & + & 2xy & - & 2y^2 \\ \hline 5x^2 & - & 6xy & - & 3y^2 \\ \hline \end{array}$

Sum 2 = 5x² - 6xy - 3y²

Let's subtract sum 1 from sum 2:

$\begin{array}{rcccc} 5x^2 & - & 6xy & - & 3y^2 \\ -2x^2 & - & 7xy & + & 3y^2 \\ +\phantom{2x^2} & + & & - \\ \hline 7x^2 & + & xy & - & 6y^2 \\ \hline \end{array}$

Hence, the answer is 7x² + xy - 6y²

(i) What should be subtracted from x + 2y - 3z to get 3x - 2y + z?

(ii) What should be subtracted from 2x² - y² + 4z² to get x² + y² - z²?

(iii) What should be subtracted from 1 + x - x² to get 2x + x²?

Answer

(i) What should be subtracted from x + 2y - 3z to get 3x - 2y + z?

To find what must be subtracted from A to get B, we simply calculate A - B.

∴ Subtract 3x - 2y + z from x + 2y - 3z

$\begin{array}{rcccc} x & + & 2y & - & 3z \\ +3x & - & 2y & + & z \\ -\phantom{3x} & + & & - \\ \hline -2x & + & 4y & - & 4z \\ \hline \end{array}$

∴ The required expression is -2x + 4y - 4z

(ii) What should be subtracted from 2x² - y² + 4z² to get x² + y² - z²?

To find what must be subtracted from A to get B, we simply calculate A - B.

∴ Subtract x² + y² - z² from 2x² - y² + 4z²

$\begin{array}{rcccc} 2x^2 & - & y^2 & + & 4z^2 \\ +x^2 & + & y^2 & - & z^2 \\ -\phantom{x^2} & - & & + \\ \hline x^2 & - & 2y^2 & + & 5z^2 \\ \hline \end{array}$

∴ The required expression is x² - 2y² + 5z²

(iii) What should be subtracted from 1 + x - x² to get 2x + x²?

To find what must be subtracted from A to get B, we simply calculate A - B.

∴ Subtract 2x + x² from 1 + x - x²

Arranging the expressions in ascending powers (starting with the constant) and use 0 as a placeholder for missing terms.

$\begin{array}{rcccc} 1 & + & x & - & x^2 \\ +0 & + & 2x & + & x^2 \\ -\phantom{0} & - & & - \\ \hline 1 & - & x & - & 2x^2 \\ \hline \end{array}$

∴ The required expression is 1 - x - 2x²

(i) What should be added to 7a - 9b + 13c to get 9a + b - c?

(ii) What should be added to 1 + 2x - 3x² to get x² - x - 1?

(iii) What should be added to m² - 2mn + 5n² to get n² + mn - m²?

Answer

(i) What should be added to 7a - 9b + 13c to get 9a + b - c?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract 7a - 9b + 13c from 9a + b - c:

$\begin{array}{rcccc} 9a & + & b & - & c \\ +7a & - & 9b & + & 13c \\ -\phantom{7a} & + & & - \\ \hline 2a & + & 10b & - & 14c \\ \hline \end{array}$

∴ The required expression is 2a + 10b - 14c

(ii) What should be added to 1 + 2x - 3x² to get x² - x - 1?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract 1 + 2x - 3x² from x² - x - 1:

Arranging the expressions in descending powers of x (x², x, constant).

$\begin{array}{rcccc} x^2 & - & x & - & 1 \\ -3x^2 & + & 2x & + & 1 \\ +\phantom{3x^2} & - & & - \\ \hline 4x^2 & - & 3x & - & 2 \\ \hline \end{array}$

∴ The required expression is 4x² - 3x - 2

(iii) What should be added to m² - 2mn + 5n² to get n² + mn - m²?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract m² - 2mn + 5n² from n² + mn - m²

Arranging the terms to match (m², mn, n²):

$\begin{array}{rcccc} -m^2 & + & mn & + & n^2 \\ +m^2 & - & 2mn & + & 5n^2 \\ -\phantom{m^2} & + & & - \\ \hline -2m^2 & + & 3mn & - & 4n^2 \\ \hline \end{array}$

∴ The required expression is -2m² + 3mn - 4n²

Find the excess of 4p² - 2pq + 3q² over 2p² - pq + 4q².

Answer

"Excess" means how much larger the first expression is than the second. To find it, we subtract the second from the first (A - B).

∴ Subtract 2p² - pq + 4q² from 4p² - 2pq + 3q².

$\begin{array}{rcccc} 4p^2 & - & 2pq & + & 3q^2 \\ +2p^2 & - & pq & + & 4q^2 \\ -\phantom{2p^2} & + & & - \\ \hline 2p^2 & - & pq & - & q^2 \\ \hline \end{array}$

Hence, the answer is 2p² - pq - q²

By how much does 3x³ - 5x² + 2x - 3 exceed 2x³ - 3x² + x + 1?

Answer

"Exceed" is just another way of asking for the difference. Subtract the second expression from the first.

∴ Subtract 2x³ - 3x² + x + 1 from 3x³ - 5x² + 2x - 3

$\begin{array}{rcccccc} 3x^3 & - & 5x^2 & + & 2x & - & 3 \\ +2x^3 & - & 3x^2 & + & x & + & 1 \\ -\phantom{2x^3} & + & - & - \\ \hline x^3 & - & 2x^2 & + & x & - & 4 \\ \hline \end{array}$

Hence, the answer is x³ - 2x² + x - 4

How much is -x² + 7y² - 3xy less than 2x² - y² + xy?

Answer

Here, "how much is A less than B," means B is the larger value. Therefore, we subtract A from B (B - A).

∴ Subtract -x² + 7y² - 3xy from 2x² - y² + xy

$\begin{array}{rcc} 2x^2 & - & y^2 & + & xy \\ -x^2 & + & 7y^2 & - & 3xy \\ +\phantom{x^2} & - & & + \\ \hline 3x^2 & - & 8y^2 & + & 4xy \\ \hline \end{array}$

Hence, the answer is 3x² - 8y² + 4xy

How much is x³ - 3x² + 5x - 2 less than 3 - 2x + x² - x³?

Answer

Here it asks "how much is A less than B," it means B is the larger value. Therefore, we subtract A from B (B - A).

∴ Subtract (x³ - 3x² + 5x - 2) from (3 - 2x + x² - x³)

Arranging the expressions in ascending powers of x (constant, x, x², x³).

$\begin{array}{rcccccc} 3 & - & 2x & + & x^2 & - & x^3 \\ -2 & + & 5x & - & 3x^2 & + & x^3 \\ +\phantom{2} & - & & + & & - \\ \hline 5 & - & 7x & + & 4x^2 & - & 2x^3 \\ \hline \end{array}$

Hence, the answer is 5 - 7x + 4x² - 2x³

If x = 2a² + 3b² - 5ab, y = b² - 3a² + 7ab and z = 6a² - b² + ab, find :

(i) x + y - z

(ii) x - y + z

Answer

Given:

x = 2a² + 3b² - 5ab

y = b² - 3a² + 7ab

z = 6a² - b² + ab

(i) x + y - z

First, let's calculate x + y:

Arranging the terms to match (ab, b², a²):

$\begin{array}{rcccc} -5ab & + & 3b^2 & + & 2a^2 \\ +7ab & + & b^2 & - & 3a^2 \\ \hline 2ab & + & 4b^2 & - & a^2 \\ \hline \end{array}$

The sum is 2ab + 4b² - a² .

Now, subtract z from the above sum:

$\begin{array}{rcccc} 2ab & + & 4b^2 & - & a^2 \\ +ab & - & b^2 & + & 6a^2 \\ -\phantom{ab} & + & & - \\ \hline ab & + & 5b^2 & - & 7a^2 \\ \hline \end{array}$

∴ x + y - z = ab + 5b² - 7a²

(ii) x - y + z

First, let's calculate x - y:

Arranging the terms to match (a², b², ab):

$\begin{array}{rcccc} 2a^2 & + & 3b^2 & - & 5ab \\ -3a^2 & + & b^2 & + & 7ab \\ +\phantom{3a^2} & - & & - \\ \hline 5a^2 & + & 2b^2 & - & 12ab \\ \hline \end{array}$

The result is 5a² + 2b² - 12ab.

Now, add z to the above result:

$\begin{array}{rcccc} 5a^2 & + & 2b^2 & - & 12ab \\ +6a^2 & - & b^2 & + & ab \\ \hline 11a^2 & + & b^2 & - & 11ab \\ \hline \end{array}$

∴ x - y + z = 11a² + b² - 11ab

The perimeter of a triangle is 8 + 13a + 7a² and two of its sides are 2a² + 3a + 2 and 3a² - 4a - 1. Find the third side of the triangle.

Answer

Given:

Perimeter of a triangle = 8 + 13a + 7a²

Side 1 (S1) = 2a² + 3a + 2

Side 2 (S2) = 3a² - 4a - 1

Side 3 (S3) = ?

We know the formula,

Perimeter of a triangle = S1 + S2 + S3

⟹ S3 = Perimeter of a triangle - (S1 + S2)

Substituting the values above, we get:

S3 = (8 + 13a + 7a²) - ((2a² + 3a + 2) + (3a² - 4a - 1))

First, let's find S1 + S2:

$\begin{array}{rcccc} 2 & + & 3a & + & 2a^2 \\ -1 & - & 4a & + & 3a^2 \\ \hline 1 & - & a & + & 5a^2 \\ \hline \end{array}$

Sum (S1 + S2) = 1 - a + 5a²

Now, we have S3 = (8 + 13a + 7a²) - (1 - a + 5a²)

Subtract Sum (S1 + S2) from perimeter of a triangle:

$\begin{array}{rcccc} 8 & + & 13a & + & 7a^2 \\ +1 & - & a & + & 5a^2 \\ -\phantom{1} & + & & - \\ \hline 7 & + & 14a & + & 2a^2 \\ \hline \end{array}$

S3 = 7 + 14a + 2a²

∴ The third side of the triangle is 7 + 14a + 2a².

The perimeter of a rectangle is 16x³ - 6x² + 12x + 4. If one of its sides is 8x² + 3x, find the other side.

Answer

Given:

Perimeter of a rectangle = 16x³ - 6x² + 12x + 4

Length (Known side) = 8x² + 3x

Breadth = ?

We know the formula,

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

Multiply Length by 2:

2(Length) = 2(8x² + 3x) = 16x² + 6x

Now, subtract 2(Length) from Perimeter:

$\begin{array}{rcccccc} 16x^3 & - & 6x^2 & + & 12x & + & 4 \\ +0 & + & 16x^2 & + & 6x & + & 0 \\ -\phantom{0} & - & & - & & - \\ \hline 16x^3 & - & 22x^2 & + & 6x & + & 4 \\ \hline \end{array}$

Now we have:

Breadth = $\dfrac{16x^3 - 22x^2 + 6x + 4}{2}$ = 8x³ - 11x² + 3x + 2

∴ The other side of the rectangle is 8x³ - 11x² + 3x + 2.