Examkin

Question

Difficulty: HardSimplifying Expressions and Combining Like Terms

If xx and yy are real numbers, the expression 3x(x−2y)2−x2(3x−10y)−y2(2x−y)3x(x - 2y)^2 - x^2(3x - 10y) - y^2(2x - y) can be simplified to which of the following?

  1. A
    −22x2y+10xy2−y3-22x^2y + 10xy^2 - y^3
  2. B
    −2x2y+10xy2−y3-2x^2y + 10xy^2 - y^3
  3. −2x2y+10xy2+y3-2x^2y + 10xy^2 + y^3Answer
  4. D
    10x2y+10xy2+y310x^2y + 10xy^2 + y^3
  5. E
    −2x2y+10xy2+y2-2x^2y + 10xy^2 + y^2

Answer

The correct expression is −2x2y+10xy2+y3-2x^2y + 10xy^2 + y^3.
The correct expression −2x2y+10xy2+y3-2x^2y + 10xy^2 + y^3 is obtained by correctly expanding the binomial (x−2y)2(x-2y)^2 into x2−4xy+4y2x^2 - 4xy + 4y^2, distributing the outer terms 3x3x, −x2-x^2, and −y2-y^2 to all terms inside their respective parentheses, and then combining the coefficients of the matching variable terms: the x3x^3 terms cancel out, the x2yx^2y terms combine to −2x2y-2x^2y, the xy2xy^2 terms combine to 10xy210xy^2, and the y3y^3 term remains.

Step-by-Step Solution

1
Expand the binomial expression (x−2y)2(x - 2y)^2.
x2−4xy+4y2x^2 - 4xy + 4y^2
Before distributing 3x3x, the squared binomial must be expanded using the identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
2
Multiply the expanded binomial by the coefficient 3x3x.
3x3−12x2y+12xy23x^3 - 12x^2y + 12xy^2
Distribute 3x3x to each of the three terms in the expanded expression: 3x(x2)=3x33x(x^2) = 3x^3, 3x(−4xy)=−12x2y3x(-4xy) = -12x^2y, and 3x(4y2)=12xy23x(4y^2) = 12xy^2.
3
Distribute −x2-x^2 to (3x−10y)(3x - 10y).
−3x3+10x2y-3x^3 + 10x^2y
Distribute −x2-x^2 to both terms, noting that multiplying two negative signs yields a positive term: −x2(3x)=−3x3-x^2(3x) = -3x^3 and −x2(−10y)=+10x2y-x^2(-10y) = +10x^2y.
4
Distribute −y2-y^2 to (2x−y)(2x - y).
−2xy2+y3-2xy^2 + y^3
Distribute −y2-y^2 to both terms, adding the exponents of yy where appropriate: −y2(2x)=−2xy2-y^2(2x) = -2xy^2 and −y2(−y1)=+y3-y^2(-y^1) = +y^3.
5
Combine all parts and group the like terms together.
−2x2y+10xy2+y3-2x^2y + 10xy^2 + y^3
Group and add the coefficients of identical variable terms: (3x3−3x3)+(−12x2y+10x2y)+(12xy2−2xy2)+y3=0x3−2x2y+10xy2+y3(3x^3 - 3x^3) + (-12x^2y + 10x^2y) + (12xy^2 - 2xy^2) + y^3 = 0x^3 - 2x^2y + 10xy^2 + y^3.

Key Concept

Simplifying polynomial expressions with multiple variables by expanding binomials, distributing coefficients (including negative signs), and combining like terms.
Rate this question

Topics

Simplifying Expressions and Combining Like Terms

More questions from this topic