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Difficulty: MediumMultiplying Polynomials

Which of the following represents the fully expanded and simplified form of the polynomial expression (2x−3)(x2−4x+5)(2x - 3)(x^2 - 4x + 5)?

  1. 2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15Answer
  2. B
    2x3−5x2−2x−152x^3 - 5x^2 - 2x - 15
  3. C
    2x3−152x^3 - 15
  4. D
    2x3+11x−152x^3 + 11x - 15

Answer

The simplified polynomial is 2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15.
The correct answer is found by distributing each term in the binomial to each term in the trinomial and combining like terms. Distributing 2x2x yields 2x3−8x2+10x2x^3 - 8x^2 + 10x, and distributing −3-3 yields −3x2+12x−15-3x^2 + 12x - 15. Combining −8x2-8x^2 with −3x2-3x^2 gives −11x2-11x^2, and combining 10x10x with 12x12x gives 22x22x, resulting in the final simplified expression 2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15.

Step-by-Step Solution

1
Distribute the first term of the binomial, 2x2x, to each term in the trinomial (x2−4x+5)(x^2 - 4x + 5).
2x(x2)+2x(−4x)+2x(5)=2x3−8x2+10x2x(x^2) + 2x(-4x) + 2x(5) = 2x^3 - 8x^2 + 10x
To multiply polynomials, every term in the first polynomial must be multiplied by every term in the second.
2
Distribute the second term of the binomial, −3-3, to each term in the trinomial.
−3(x2)−3(−4x)−3(5)=−3x2+12x−15-3(x^2) - 3(-4x) - 3(5) = -3x^2 + 12x - 15
It is critical to carry the negative sign when distributing to ensure the signs of the resulting terms are correct.
3
Add the results from both distribution steps and group the like terms together.
2x3+(−8x2−3x2)+(10x+12x)−152x^3 + (-8x^2 - 3x^2) + (10x + 12x) - 15
Grouping terms with the same variable and exponent makes it easier to simplify.
4
Combine the coefficients of the grouped like terms.
2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15
Combining like terms provides the expression in its fully simplified, standard polynomial form.

Key Concept

Multiplying Polynomials
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Topics

Multiplying Polynomials

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