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Question

Difficulty: MediumMultiplying Polynomials

When the expression (3x−4)(2x+5)−x(x−2)(3x - 4)(2x + 5) - x(x - 2) is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of xx?

Answer: 9

Answer

9
Expanding the binomials gives 6x2+7x−206x^2 + 7x - 20. Distributing −x-x gives −x2+2x-x^2 + 2x. Combining like terms results in 5x2+9x−205x^2 + 9x - 20. The question asks for the coefficient of xx, which is the number multiplying the xx term: 9.

Step-by-Step Solution

1
Multiply the binomials (3x−4)(3x - 4) and (2x+5)(2x + 5).
6x2+15x−8x−206x^2 + 15x - 8x - 20, which simplifies to 6x2+7x−206x^2 + 7x - 20.
To expand the first part of the expression using the distributive property (FOIL).
2
Distribute −x-x across the binomial (x−2)(x - 2).
−x2+2x-x^2 + 2x.
To expand the second part of the expression, ensuring the negative sign is applied to both terms inside the parentheses.
3
Combine the expanded parts.
(6x2+7x−20)+(−x2+2x)=5x2+9x−20(6x^2 + 7x - 20) + (-x^2 + 2x) = 5x^2 + 9x - 20.
To simplify the full expression into standard polynomial form by grouping like terms.
4
Identify the coefficient of xx.
The coefficient of xx in 5x2+9x−205x^2 + 9x - 20 is 99.
This answers the specific question asked.

Key Concept

Multiplying polynomials and combining like terms
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Topics

Multiplying Polynomials

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