Question

polynomials review

1. please simplify the following.

a) (3x - 10y + 4y - 7x)
b) (14n + 23x - 17nx + 3n - 6nx)
c) (1x - 4x - (3b + 2x) - 3b)
d) (9xy^2 + 6xy - 8xy^2 - 8x^2y + 6x^2y^2 - 6xy)

1. please state the degree of each polynomial.

a) (x^2 + 2x^2y - 3xy) _______
b) (xyz - 3xy^2z + 4xy) _______

1. simplify the following.

a) ((3z - 4d + 2c) - (3z + 7d - 9c))
b) (3(2x(x - 2)) + (7 - 2(x + 1)))
c) (20x^2y^3 - 3x^2y(3x + 2y - 1))
d) ((4x + 1)(2x^2 - 3) - (3x - 2)(x^2 - 1))

Explanation:

⚡ Using: simplifying expressions · 🆕 New: degree of a polynomial
The highest total exponent sum in any single term.

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Step 1: Simplify Question 1

Group and combine like terms (terms with the exact same variables and exponents).

a) \( 3x - 10y + 4y - 7x \)

Combine \( x \)-terms and \( y \)-terms:

$$ (3x - 7x) + (-10y + 4y) = -4x - 6y $$

b) \( 14n + 23x - 17nx + 3n - 6nx \)

Combine \( n \)-terms, \( x \)-terms, and \( nx \)-terms:

$$ (14n + 3n) + 23x + (-17nx - 6nx) = 17n + 23x - 23nx $$

c) \( 1x - 4x - (3b + 2x) - 3b \)

Distribute the negative sign through the parentheses:

$$ 1x - 4x - 3b - 2x - 3b $$

Combine \( x \)-terms and \( b \)-terms:

$$ (1x - 4x - 2x) + (-3b - 3b) = -5x - 6b $$

d) \( 9xy^2 + 6xy - 8xy^2 - 8x^2y + 6x^2y^2 - 6xy \)

Combine like terms:

• \( xy^2 \) terms: \( 9xy^2 - 8xy^2 = 1xy^2 \)
• \( xy \) terms: \( 6xy - 6xy = 0 \)
• \( x^2y \) terms: \( -8x^2y \)
• \( x^2y^2 \) terms: \( 6x^2y^2 \)

$$ 6x^2y^2 - 8x^2y + xy^2 $$

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Step 2: Determine the Degree in Question 2

The degree of a polynomial is the highest sum of the exponents of the variables in any single term.

a) \( x^2 + 2x^2y - 3xy \)

• Term 1: \( x^2 \) (degree = 2)
• Term 2: \( 2x^2y^1 \) (degree = \( 2 + 1 = 3 \))
• Term 3: \( -3x^1y^1 \) (degree = \( 1 + 1 = 2 \))

The highest degree is 3.

b) \( xyz - 3xy^2z + 4xy \)

• Term 1: \( x^1y^1z^1 \) (degree = \( 1 + 1 + 1 = 3 \))
• Term 2: \( -3x^1y^2z^1 \) (degree = \( 1 + 2 + 1 = 4 \))
• Term 3: \( 4x^1y^1 \) (degree = \( 1 + 1 = 2 \))

The highest degree is 4.

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Step 3: Simplify Question 3

a) \( (3z - 4d + 2c) - (3z + 7d - 9c) \)

Distribute the negative sign:

$$ 3z - 4d + 2c - 3z - 7d + 9c $$

Combine like terms:

$$ (3z - 3z) + (-4d - 7d) + (2c + 9c) = -11d + 11c $$

b) \( 3(2x(x - 2)) + (7 - 2(x + 1)) \)

Simplify inside the brackets first:

$$ 3(2x^2 - 4x) + (7 - 2x - 2) $$

$$ (6x^2 - 12x) + (5 - 2x) $$

Combine like terms:

$$ 6x^2 - 14x + 5 $$

c) \( 20x^2y^3 - 3x^2y(3x + 2y - 1) \)

Distribute \( -3x^2y \) to each term inside the parentheses:

$$ 20x^2y^3 - 9x^3y - 6x^2y^2 + 3x^2y $$

d) \( (4x + 1)(2x^2 - 3) - (3x - 2)(x^2 - 1) \)

Expand the first part:

$$ (4x + 1)(2x^2 - 3) = 8x^3 - 12x + 2x^2 - 3 = 8x^3 + 2x^2 - 12x - 3 $$

Expand the second part:

$$ (3x - 2)(x^2 - 1) = 3x^3 - 3x - 2x^2 + 2 = 3x^3 - 2x^2 - 3x + 2 $$

Subtract the second expanded expression from the first:

$$ (8x^3 + 2x^2 - 12x - 3) - (3x^3 - 2x^2 - 3x + 2) $$

$$ 8x^3 + 2x^2 - 12x - 3 - 3x^3 + 2x^2 + 3x - 2 $$

Combine like terms:

$$ (8x^3 - 3x^3) + (2x^2 + 2x^2) + (-12x + 3x) + (-3 - 2) $$

$$ 5x^3 + 4x^2 - 9x - 5 $$

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Answer:

1) Simplify:

• a) \( -4x - 6y \)
• b) \( 17n + 23x - 23nx \)
• c) \( -5x - 6b \)
• d) \( 6x^2y^2 - 8x^2y + xy^2 \)

2) Degree of each polynomial:

• a) \( 3 \)
• b) \( 4 \)

3) Simplify:

• a) \( -11d + 11c \)
• b) \( 6x^2 - 14x + 5 \)
• c) \( 20x^2y^3 - 9x^3y - 6x^2y^2 + 3x^2y \)
• d) \( 5x^3 + 4x^2 - 9x - 5 \)