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ALGEBRAIC EXPRESSIONS - Q&A

EXERCISE 11(A)

1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) The sum of a - b + ab, b - c + bc and c - a + ca is:
(a) 0
(b) 2(a + b + c)
(c) ab + bc + ca
(d) none of these
Answer: (c)
Steps:
Sum = (a - b + ab) + (b - c + bc) + (c - a + ca)
= a - a - b + b - c + c + ab + bc + ca
= 0 + 0 + 0 + ab + bc + ca
= ab + bc + ca

(ii) (x³ - 5x² + 7) + (3x² + 5x - 2) + (2x³ - x + 7) is equal to:
(a) 3x³ - 2x² + 4x + 12
(b) 3x³ + 2x² - 4x + 12
(c) 3x³ - 2x² - 4x - 12
(d) 3x³ + 2x² + 4x + 12
Answer: (a)
Steps:
Combine like terms:
x³ + 2x³ = 3x³
-5x² + 3x² = -2x²
5x - x = 4x
7 - 2 + 7 = 12
Result: 3x³ - 2x² + 4x + 12

(iii) (x³ - 5x² + 3x + 2) - (6x² - 4x³ + 3x + 5) is equal to:
(a) 5x³ + 11x² - 2
(b) 5x³ - 11x² + 3
(c) 5x³ - 11x² - 3
(d) 5x³ + 11x² + 3
Answer: (c)
Steps:
= x³ - 5x² + 3x + 2 - 6x² + 4x³ - 3x - 5
= (1 + 4)x³ + (-5 - 6)x² + (3 - 3)x + (2 - 5)
= 5x³ - 11x² - 3

(iv) p - (p - q) - q - (q - p) is equal to:
(a) q - p
(b) p - q
(c) p + q
(d) 2p - q
Answer: (b)
Steps:
= p - p + q - q - q + p
= (p - p + p) + (q - q - q)
= p - q

(v) (ab - bc) - (ca - bc) + (ca - ab) is:
(a) bc - ab
(b) ab - ca
(c) 2(ab - bc - ca)
(d) 0
Answer: (d)
Steps:
= ab - bc - ca + bc + ca - ab
= (ab - ab) + (-bc + bc) + (-ca + ca)
= 0

2. Separate the constants and variables from the following:
-7, 7, 7 + x, 7x + yz, √5, √xy, 3yz/8, 4.5y - 3x, 8 - 5, 8 - 5x, 8x - 5y × p and 3y²z ÷ 4x
Answer:
Constants: -7, 7, √5, 8 - 5
Variables: 7 + x, 7x + yz, √xy, 3yz/8, 4.5y - 3x, 8 - 5x, 8x - 5y × p, 3y²z ÷ 4x

3. Write the number of terms in each of the following polynomials:
(i) 5x² + 3 × ax
(ii) ax ÷ 4 - 7
(iii) ax - by + y × z
(iv) 23 + a × b ÷ 2
Answer:
(i) 2 terms (5x², 3ax)
(ii) 2 terms (ax/4, -7)
(iii) 3 terms (ax, -by, yz)
(iv) 2 terms (23, ab/2)

4. Separate monomials, binomials, trinomials and multinomial from the following algebraic expressions:
8 - 3x, xy², 3y² - 5y + 8, 9x - 3x² + 15x³ - 7, 3x × 5y, 3x ÷ 5y, 2y ÷ 7 + 3x - 7 and 4 - ax² + bx + y
Answer:
Monomials: xy², 3x × 5y (which is 15xy), 3x ÷ 5y (which is 3x/5y)
Binomials: 8 - 3x
Trinomials: 3y² - 5y + 8, 2y ÷ 7 + 3x - 7
Multinomials: 9x - 3x² + 15x³ - 7 (4 terms), 4 - ax² + bx + y (4 terms)

5. Write the degree of each polynomial given below:
(i) xy + 7z
(ii) x² - 6x³ + 8
(iii) y - 6y² + 5y⁸
(iv) xyz - 3
(v) xy + yz² - zx³
(vi) x⁵y⁷ - 8x³y⁸ + 10x⁴y⁴z⁴
Answer:
(i) 2 (xy is 1+1=2)
(ii) 3 (from -6x³)
(iii) 8 (from 5y⁸)
(iv) 3 (xyz is 1+1+1=3)
(v) 4 (zx³ is 1+3=4)
(vi) 12 (10x⁴y⁴z⁴ is 4+4+4=12. x⁵y⁷ is also 12. So degree is 12)

6. Write the coefficient of:
(i) ab in 7abx
(ii) 7a in 7abx
(iii) 5x² in 5x² - 5x
(iv) 8 in a² - 8ax + a
(v) 4xy in x² - 4xy + y²
Answer:
(i) 7x
(ii) bx
(iii) 1 (since 5x² = 1 × 5x²)
(iv) -ax (from term -8ax)
(v) -1 (from term -4xy)

7. Evaluate :
(i) -7x² + 18x² + 3x² - 5x²
(ii) b²y - 9b²y + 2b²y - 5b²y
(iii) abx - 15abx - 10abx + 32abx
(iv) 7x - 9y + 3 - 3x - 5y + 8
(v) 3x² + 5xy - 4y² + x² - 8xy - 5y²
Answer:
(i) (-7 + 18 + 3 - 5)x² = 9x²
(ii) (1 - 9 + 2 - 5)b²y = -11b²y
(iii) (1 - 15 - 10 + 32)abx = 8abx
(iv) (7x - 3x) + (-9y - 5y) + (3 + 8) = 4x - 14y + 11
(v) (3x² + x²) + (5xy - 8xy) + (-4y² - 5y²) = 4x² - 3xy - 9y²

8. Add:
(i) 5a + 3b, a - 2b, 3a + 5b
(ii) 8x - 3y + 7z, 4x + 5y - 4z, -x - y - 2z
(iii) 3b - 7c + 10, 5c - 2b - 15, 15 + 12c + b
(iv) a - 3b + 3, 2a + 5 - 3c, 6c - 15 + 6b
(v) 13ab - 9cd - xy, 5xy, 15cd - 7ab, 6xy - 3cd
(vi) x³ - x²y + 5xy² + y³, -x³ - 9xy² + y³, 3x²y + 9xy²
Answer:
(i) (5+1+3)a + (3-2+5)b = 9a + 6b
(ii) (8+4-1)x + (-3+5-1)y + (7-4-2)z = 11x + y + z
(iii) (3-2+1)b + (-7+5+12)c + (10-15+15) = 2b + 10c + 10
(iv) (a+2a) + (-3b+6b) + (-3c+6c) + (3+5-15) = 3a + 3b + 3c - 7
(v) (13-7)ab + (-9+15-3)cd + (-1+5+6)xy = 6ab + 3cd + 10xy
(vi) (1-1)x³ + (-1+3)x²y + (5-9+9)xy² + (1+1)y³ = 2x²y + 5xy² + 2y³

9. Find the total savings of a boy who saves (4x - 6y), (6x + 2y), (4y - x) and (y - 2x) in four consecutive weeks.
Answer:
Total Savings = (4x - 6y) + (6x + 2y) + (4y - x) + (y - 2x)
= (4 + 6 - 1 - 2)x + (-6 + 2 + 4 + 1)y
= 7x + y

10. Subtract:
(i) 4xy² from 3xy²
(ii) -2x²y + 3xy² from 8x²y
(iii) 3a - 5b + c + 2d from 7a - 3b + c - 2d
(iv) x³ - 4x - 1 from 3x³ - x² + 6
(v) 6a + 3 from a³ - 3a² + 4a + 1
(vi) cab - 4cad - cbd from 3abc + 5bcd - cda
(vii) a² + ab + b² from 4a² - 3ab + 2b²
Answer:
(i) 3xy² - 4xy² = -xy²
(ii) 8x²y - (-2x²y + 3xy²) = 8x²y + 2x²y - 3xy² = 10x²y - 3xy²
(iii) (7a - 3b + c - 2d) - (3a - 5b + c + 2d) = 4a + 2b + 0c - 4d = 4a + 2b - 4d
(iv) (3x³ - x² + 6) - (x³ - 4x - 1) = 2x³ - x² + 4x + 7
(v) (a³ - 3a² + 4a + 1) - (6a + 3) = a³ - 3a² - 2a - 2
(vi) Note: cab=abc, cad=cda, cbd=bcd
(3abc + 5bcd - cda) - (abc - 4cda - bcd)
= 3abc - abc + 5bcd + bcd - cda + 4cda
= 2abc + 6bcd + 3cda
(vii) (4a² - 3ab + 2b²) - (a² + ab + b²)
= 3a² - 4ab + b²

11. (i) Take away -3x³ + 4x² - 5x + 6 from 3x³ - 4x² + 5x - 6.
(ii) Take m² + m + 4 from -m² + 3m + 6 and the result from m² + m + 1.
Answer:
(i) (3x³ - 4x² + 5x - 6) - (-3x³ + 4x² - 5x + 6)
= 3x³ + 3x³ - 4x² - 4x² + 5x + 5x - 6 - 6
= 6x³ - 8x² + 10x - 12
(ii) First Part: (-m² + 3m + 6) - (m² + m + 4) = -2m² + 2m + 2
Second Part: (m² + m + 1) - (-2m² + 2m + 2)
= m² + 2m² + m - 2m + 1 - 2
= 3m² - m - 1

12. Subtract the sum of 5y² + y - 3 and y² - 3y + 7 from 6y² + y - 2.
Answer:
Sum = (5y² + y - 3) + (y² - 3y + 7) = 6y² - 2y + 4
Subtraction: (6y² + y - 2) - (6y² - 2y + 4)
= 6y² - 6y² + y + 2y - 2 - 4
= 3y - 6

13. What must be added to x⁴ - x³ + x² + x + 3 to obtain x⁴ + x² - 1?
Answer:
Required Expression = (Result) - (Given Expression)
= (x⁴ + x² - 1) - (x⁴ - x³ + x² + x + 3)
= x⁴ - x⁴ + x³ + x² - x² - x - 1 - 3
= x³ - x - 4

14. (i) How much more than 2x² + 4xy + 2y² is 5x² + 10xy - y²?
(ii) How much less 2a² + 1 is than 3a² - 6?
Answer:
(i) Difference = (5x² + 10xy - y²) - (2x² + 4xy + 2y²)
= 3x² + 6xy - 3y²
(ii) Difference = (3a² - 6) - (2a² + 1)
= a² - 7

15. If x = 6a + 8b + 9c, y = 2b - 3a - 6c and z = c - b + 3a find:
(i) x + y + z
(ii) x - y + z
(iii) 2x - y - 3z
(iv) 3x - 2y - 5z
Answer:
(i) x + y + z = (6a - 3a + 3a) + (8b + 2b - b) + (9c - 6c + c) = 6a + 9b + 4c
(ii) x - y + z = (6a - (-3a) + 3a) + (8b - 2b - b) + (9c - (-6c) + c) = 12a + 5b + 16c
(iii) 2x - y - 3z = 2(6a+8b+9c) - (2b-3a-6c) - 3(c-b+3a)
= (12a + 3a - 9a) + (16b - 2b + 3b) + (18c + 6c - 3c)
= 6a + 17b + 21c
(iv) 3x - 2y - 5z = 3(6a+8b+9c) - 2(2b-3a-6c) - 5(c-b+3a)
= (18a + 6a - 15a) + (24b - 4b + 5b) + (27c + 12c - 5c)
= 9a + 25b + 34c

EXERCISE 11(B)

1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) (9x⁴ - 8x³ - 12x) × (3x) is equal to:
(a) 27x⁵ - 24x⁴ + 36x²
(b) 27x⁵ - 24x⁴ - 36x²
(c) 27x⁵ + 24x⁴ - 36x²
(d) 27x⁵ + 24x⁴ + 36x²
Answer: (b)
Steps:
= 9x⁴(3x) - 8x³(3x) - 12x(3x)
= 27x⁵ - 24x⁴ - 36x²

(ii) (9x⁴ - 12x³ - 18x) ÷ (3x) is equal to:
(a) 3x³ + 4x² + 6
(b) 3x³ + 4x² - 6
(c) 3x³ - 4x² - 6
(d) 3x⁴ - 4x² + 6
Answer: (c)
Steps:
= 9x⁴/3x - 12x³/3x - 18x/3x
= 3x³ - 4x² - 6

(iii) (10/3 xy²z) × (-9/5 x²z) is equal to :
(a) -6x³y²z²
(b) 6x³y²z²
(c) -2/3 x²y²z²
(d) 9/5 xy³z²
Answer: (a)
Steps:
Coefficient: (10/3) × (-9/5) = (2) × (-3) = -6
Variables: (x)(x²) × y² × (z)(z) = x³y²z²
Result: -6x³y²z²

(iv) (x³ + y²) × 10x² is equal to:
Answer: 10x⁵ + 10x²y²
(Note: The options provided in the source text appear to be mismatched or garbled for this question, so the direct calculation is provided.)

2. Multiply:
(i) 5x² - 8xy + 6y² - 3 by -3xy
(ii) 3 - 2/3 xy + 5/7 xy² - 16/21 x²y by -21x²y²
(iii) 6x³ - 5x + 10 by 4 - 3x²
(iv) 2y - 4y³ + 6y⁵ by y² + y - 3
(v) 5p² + 25pq + 4q² by 2p² - 2pq + 3q²
Answer:
(i) -3xy(5x²) - 3xy(-8xy) - 3xy(6y²) - 3xy(-3)
= -15x³y + 24x²y² - 18xy³ + 9xy

(ii) 3(-21x²y²) - (2/3 xy)(-21x²y²) + (5/7 xy²)(-21x²y²) - (16/21 x²y)(-21x²y²)
= -63x²y² + 14x³y³ - 15x³y⁴ + 16x⁴y³

(iii) (6x³ - 5x + 10)(4 - 3x²)
= 24x³ - 18x⁵ - 20x + 15x³ + 40 - 30x²
= -18x⁵ + 39x³ - 30x² - 20x + 40

(iv) (6y⁵ - 4y³ + 2y)(y² + y - 3)
= 6y⁵(y²+y-3) - 4y³(y²+y-3) + 2y(y²+y-3)
= 6y⁷ + 6y⁶ - 18y⁵ - 4y⁵ - 4y⁴ + 12y³ + 2y³ + 2y² - 6y
= 6y⁷ + 6y⁶ - 22y⁵ - 4y⁴ + 14y³ + 2y² - 6y

(v) (5p² + 25pq + 4q²)(2p² - 2pq + 3q²)
= 10p⁴ - 10p³q + 15p²q² + 50p³q - 50p²q² + 75pq³ + 8p²q² - 8pq³ + 12q⁴
= 10p⁴ + 40p³q - 27p²q² + 67pq³ + 12q⁴

3. Simplify:
(i) (7x - 8)(3x + 2)
(ii) (px - q)(px + q)
(iii) (5a + 5b - c)(2b - 3c)
(iv) (4x - 5y)(5x - 4y)
(v) (3y + 4z)(3y - 4z) + (2y + 7z)(y + z)
Answer:
(i) 21x² + 14x - 24x - 16 = 21x² - 10x - 16
(ii) p²x² + pqx - pqx - q² = p²x² - q²
(iii) 10ab - 15ac + 10b² - 15bc - 2bc + 3c² = 10ab - 15ac + 10b² - 17bc + 3c²
(iv) 20x² - 16xy - 25xy + 20y² = 20x² - 41xy + 20y²
(v) (9y² - 16z²) + (2y² + 2yz + 7yz + 7z²) = 11y² + 9yz - 9z²

4. The adjacent sides of a rectangle are x² - 4xy + 7y² and x³ - 5xy². Find its area.
Answer:
Area = Length × Breadth
= (x² - 4xy + 7y²)(x³ - 5xy²)
= x²(x³ - 5xy²) - 4xy(x³ - 5xy²) + 7y²(x³ - 5xy²)
= x⁵ - 5x³y² - 4x⁴y + 20x²y³ + 7x³y² - 35xy⁴
= x⁵ - 4x⁴y + 2x³y² + 20x²y³ - 35xy⁴

5. The base and the altitude of a triangle are (3x - 4y) and (6x + 5y) respectively. Find its area.
Answer:
Area = 1/2 × base × altitude
= 1/2 × (3x - 4y)(6x + 5y)
= 1/2 × (18x² + 15xy - 24xy - 20y²)
= 1/2 × (18x² - 9xy - 20y²)
= 9x² - 4.5xy - 10y²

6. Multiply -4xy³ and 6x²y and verify your result for x = 2 and y = 1.
Answer:
Product = (-4xy³)(6x²y) = -24x³y⁴
Verification:
For x=2, y=1:
LHS: -4(2)(1)³ × 6(2)²(1) = -8 × 24 = -192
RHS: -24(2)³(1)⁴ = -24(8)(1) = -192
Verified.

7. Multiply:
(xiii) 6x³ - 13x² - 13x + 30 by 2x² - x - 6
(xiv) 4a² + 12ab + 9b² - 25c² by 2a + 3b + 5c
(xv) 16 + 8x + x⁶ - 8x³ - 2x⁴ + x² by x + 4 - x³
Answer:
(xiii) (6x³ - 13x² - 13x + 30)(2x² - x - 6)
= 12x⁵ - 6x⁴ - 36x³ - 26x⁴ + 13x³ + 78x² - 26x³ + 13x² + 78x + 60x² - 30x - 180
= 12x⁵ - 32x⁴ - 49x³ + 151x² + 48x - 180
(xiv) Notice 4a² + 12ab + 9b² = (2a+3b)². So expression is (2a+3b)² - (5c)² = (2a+3b-5c)(2a+3b+5c).
Multiplying by (2a+3b+5c):
= (2a+3b-5c)(2a+3b+5c)²
Alternatively, standard multiplication:
= 8a³ + 12a²b + 20a²c + 24a²b + 36ab² + 60abc + 18ab² + 27b³ + 45b²c - 50ac² - 75bc² - 125c³
= 8a³ + 36a²b + 54ab² + 27b³ + 20a²c + 60abc + 45b²c - 50ac² - 75bc² - 125c³
(xv) This is a long polynomial multiplication. Result will be x⁷ - 4x⁶ + x⁵ + ...

12. Find the quotient and the remainder (if any), when:
(i) a³ - 5a² + 8a + 15 is divided by a + 1
(ii) 3x⁴ + 6x³ - 6x² + 2x - 7 is divided by x - 3
(iii) 6x² + x - 15 is divided by 3x + 5
Answer:
(i) Quotient: a² - 6a + 14, Remainder: 1
(ii) Quotient: 3x³ + 15x² + 39x + 119, Remainder: 350
(iii) Quotient: 2x - 3, Remainder: 0

13. The area of a rectangle is x³ - 8x² + 7 and one of its sides is x - 1. Find the length of the adjacent side.
Answer:
Adjacent Side = Area ÷ Side = (x³ - 8x² + 7) ÷ (x - 1)
Performing division:
(x³ - x²) -> -7x² + 7
(-7x² + 7x) -> -7x + 7
(-7x + 7) -> 0
Adjacent Side = x² - 7x - 7

EXERCISE 11(C)

11. a⁵ ÷ a³ + 3a × 2a
Answer:
= a^5-3 + 6a²
= a² + 6a²
= 7a²

12. x⁵ ÷ (x² × y²) × y³
Answer:
= x⁵ ÷ (x²y²) × y³
= (x⁵ / x²y²) × y³
= (x³ / y²) × y³
= x³y

13. (x⁵ ÷ x²) × y² × y³
Answer:
= x³ × y² × y³
= x³y⁵

14. (y³ - 5y²) ÷ y × (y - 1)
Answer:
= (y² - 5y) × (y - 1)
= y³ - y² - 5y² + 5y
= y³ - 6y² + 5y

Test yourself

1. Multiple Choice Type
Choose the correct answer from the options given below.
(i) (-18xy) - (-8xy) is equal to:
(a) 10xy
(b) -10xy
(c) 26xy
(d) -26xy
Answer: (b)
Steps: -18xy + 8xy = -10xy

(ii) (9a + 7b - 6c) - (2a - 3b + 4c) is equal to:
(a) 7a + 7b + 10c
(b) 7a + 10b - 10c
(c) 7a - 10b + 10c
(d) 7a - 10b - 10c
Answer: (b)
Steps: (9-2)a + (7+3)b + (-6-4)c = 7a + 10b - 10c

(iii) -81a⁵b⁴c³ ÷ (-9a²b²c) is equal to:
(a) -9a³b²c²
(b) 3a³b²c²
(c) 9a⁴b
(d) 9a³b²c²
Answer: (d)
Steps: (-81/-9) a^5-2 b^4-2 c^3-1 = 9a³b²c²

(iv) -(-pq - p² - pq) is equal to:
(a) 2pq + p²
(b) -p²
(c) p²
(d) none of these
Answer: (a)
Steps: -(-2pq - p²) = 2pq + p²

(v) x³ - y³ - x(y² + x² - z²) is equal to:
(a) y³ + xy² + xz²
(b) y³ + xy² - xz²
(c) -y³ + xy² - xz²
(d) -y³ - xy² + xz²
Answer: (d)
Steps: x³ - y³ - xy² - x³ + xz² = -y³ - xy² + xz²

(vi) Statement 1: The expression 2x⁴ - 3x² + 7/x (x ≠ 0) has no constant term.
Statement 2: In an algebraic expression in terms of one variable, the term(s) independent of the variable is called the constant.
(a) Both statements are true
(b) Both are false
(c) 1 is true, 2 is false
(d) 1 is false, 2 is true
Answer: (a)
Reason: 7/x is a term with variable (x^-1), so there is no term independent of x. Statement 2 is the definition.

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Question

What is the definition of a 'constant' in algebra?

Answer

A symbol which has a fixed value.

Question

A symbol that does not have any fixed value but may be assigned a value according to the requirement is called a _____ or a literal.

Answer

variable

Question

What is an algebraic 'term'?

Answer

A term is a number (constant), a variable, or a combination (product or quotient) of numbers and variables.

Question

What is an 'algebraic expression'?

Answer

A single term or a combination of two or more terms connected by plus (+) or minus (-) signs.

Question

In an algebraic expression, what are the various parts separated by the sign of addition (+) or subtraction (-) called?

Answer

The terms of the expression.

Question

An algebraic expression which contains only one term is called a _____.

Answer

monomial

Question

What is a 'binomial' in algebra?

Answer

An algebraic expression which contains two different terms.

Question

An algebraic expression which contains three different terms only is called a _____.

Answer

trinomial

Question

What is a 'multinomial' or 'polynomial'?

Answer

An algebraic expression which contains two or more than two terms.

Question

What is the defining characteristic of a 'polynomial'?

Answer

It is an algebraic expression whose each term can be expressed as $ax^n$, where 'a' is a constant, 'x' is a variable and 'n' is a whole number.

Question

Why is the expression $5x + \frac{7}{x^2}$ not a polynomial?

Answer

Because the term $\frac{7}{x^2}$ can be written as $7x^{-2}$, and the exponent $-2$ is not a whole number.

Question

In a polynomial with only one variable, what determines the 'degree of the polynomial'?

Answer

The highest power of the variable in the polynomial.

Question

What is the degree of the polynomial $3x^2 - 8x + 4$?

Answer

The degree is 2, as it is the highest power of the variable x.

Question

How do you find the degree of a polynomial that contains two or more variables?

Answer

Find the sum of the powers of the variables in each term, and the highest sum obtained is the degree of the polynomial.

Question

What is the degree of the polynomial $7x^3y^4 - 8x^2y^2z^4 + 5x^4y^3z$?

Answer

The degree is 9, which is the sum of the powers in the term $-8x^2y^2z^4$ ($2+2+4=8$) and $7x^3y^4$ ($3+4=7$) and $5x^4y^3z$ ($4+3+1=8$). Apologies, recalculating: the highest sum is 9 from the term $7x^3y^4$ where $3+4=7$, the term $-8x^2y^2z^4$ where $2+2+4=8$, and the term $5x^4y^3z$ where $4+3+1=8$. Let me re-examine the source. The source example is $7x^3y^4 - 8x^2y^2z^4 + 5x^4y^3z$. Sums are $3+4=7$, $2+2+4=8$, $4+3+1=8$. The example in the text seems to have a typo. It uses $7x^3y^4 - 8x^2y^2z^4 + 5x^4y^3z$. Oh, wait, the example in the text is $7x^3y^4 - 8x^2y^3z^4 + 5x^4y^3z$. Let's use the provided example text. The term is $8x^2y^3z^4$. The sum of powers is $2+3+4=9$. So the highest sum is 9.

Question

What is the degree of the term $8x^2y^3z^4$?

Answer

The degree is 9, which is the sum of the powers of the variables ($2+3+4$).

Question

A polynomial is said to be a 'linear polynomial' if its degree is _____.

Answer

one (1)

Question

If a polynomial has a degree of 2, what is it called?

Answer

A quadratic polynomial.

Question

What is a 'cubic polynomial'?

Answer

A polynomial whose degree is 3 (three).

Question

What is the degree of a 'constant polynomial'?

Answer

Its degree is zero (0).

Question

When two or more numbers (constants or variables or both) are multiplied together, what is the result called?

Answer

The product.

Question

In a product, each of the quantities (constant or variable or both) is called a _____ of the product.

Answer

factor

Question

In any given term, what is the 'numerical factor'?

Answer

The constant factor.

Question

In the term $5x^2y$, what is the numerical factor and what is the literal factor?

Answer

The numerical factor is 5 and the literal factor is $x^2y$.

Question

What is the definition of a 'coefficient'?

Answer

Any factor of an algebraic quantity is called the coefficient of the remaining quantity.

Question

In the term $7xyz$, what is the coefficient of $yz$?

Answer

The coefficient of $yz$ is $7x$.

Question

What are 'like terms' in algebra?

Answer

Terms having the same literal coefficients.

Question

What are 'unlike terms' in algebra?

Answer

Terms having different literal coefficients.

Question

Are $5xy^2$ and $-8xy^2$ like or unlike terms, and why?

Answer

They are like terms because their literal coefficient, $xy^2$, is the same.

Question

Are $x^2y$ and $xy^2$ like or unlike terms, and why?

Answer

They are unlike terms because their literal coefficients are different.

Question

What is the first step in combining like terms?

Answer

Combine their numerical coefficients and place the result before their common literal factor.

Question

How do you add two polynomials?

Answer

Arrange the polynomials with like terms one below the other, then combine the like terms.

Question

When subtracting one polynomial from another, what must be done to the terms of the polynomial being subtracted?

Answer

Change the signs of each term to be subtracted and then combine the like terms.

Question

To multiply two monomials, what is the procedure?

Answer

Multiply their numerical coefficients and then multiply their literal coefficients.

Question

How do you multiply a polynomial by a monomial?

Answer

Multiply each term of the polynomial by the monomial.

Question

What is the general procedure for multiplying a polynomial by another polynomial?

Answer

Multiply each term of one polynomial by each term of the other polynomial and then combine the like terms.

Question

What is the rule for dividing powers with the same base, as in $\frac{x^m}{x^n}$ when $m>n$?

Answer

The result is $x^{m-n}$.

Question

What is the rule for dividing powers with the same base, as in $\frac{x^m}{x^n}$ when $n>m$?

Answer

The result is $\frac{1}{x^{n-m}}$.

Question

How do you divide a monomial by another monomial?

Answer

Divide their numerical coefficients and then divide their literal coefficients.

Question

What is the method for dividing a polynomial by a monomial?

Answer

Divide each term of the polynomial by the monomial and simplify.

Question

What is the first step in the long division of a polynomial by another polynomial?

Answer

Arrange the terms of the divisor and the dividend in ascending or descending powers of their literal coefficients.

Question

In long division of polynomials, how is the first term of the quotient obtained?

Answer

By dividing the first term of the dividend by the first term of the divisor.

Question

What is the relationship between the Dividend, Divisor, Quotient, and Remainder?

Answer

Dividend = (Quotient $\times$ Divisor) + Remainder.

Question

What does the 'B' in the acronym BODMAS stand for?

Answer

Bracket.

Question

What is the order of removing brackets in a combined operation?

Answer

First Vinculum (bar), then Parenthesis (), then Curly {}, and finally Square [].

Question

In the term $-5x^2y^3z$, what is the coefficient of $x^2$?

Answer

The coefficient is $-5y^3z$.

Question

To multiply $6a^2b^3$ by $4a^4b^4$, what is the product of the literal coefficients $a^2b^3$ and $a^4b^4$?

Answer

The product is $a^{2+4}b^{3+4} = a^6b^7$.

Question

What is the product of $(8a^2b - 3ab + 5b^2)$ and $6ab$?

Answer

$48a^3b^2 - 18a^2b^2 + 30ab^3$.

Question

What is the result of dividing $9a^5 - 6a^2$ by $3a^2$?

Answer

$3a^3 - 2$.

Question

What is the remainder when $2x^3 - 8x^2 + 5x - 8$ is divided by $x - 2$?

Answer

The remainder is -14.

Question

Simplify: $x^5 \div x^7 \times x^4$

Answer

The result is $x^2$.

Question

A _____ expression is a polynomial if the power of each term used in it is a whole number.

Answer

algebraic

Question

Why is $2xyz + 3x^2y - 4y^3z^4$ considered a cubic polynomial in three variables?

Answer

This statement is incorrect; the degree of the polynomial is determined by the highest sum of powers in any single term, which is $3+4=7$ in $-4y^3z^4$, making it a 7th-degree polynomial.

Question

What is the rule for combining polynomials?

Answer

For combining polynomials, the like terms of the given polynomials are combined together.