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Zorluk: OrtaMultiplying Polynomials

Multiply the binomial (3x−4)(3x - 4) by the trinomial (5x2−2x−7)(5x^2 - 2x - 7). After fully expanding the product and combining all like terms, what is the numerical coefficient of the xx term?

Cevap: -13

Cevap

The coefficient of the x term is -13.
The coefficient of the xx term is found by locating all the xx terms in the expanded expression (−21x-21x and 8x8x) and adding their coefficients together (−21+8=−13-21 + 8 = -13).

Adım Adım Çözüm

1
Distribute the term 3x3x to every term in the trinomial (5x2−2x−7)(5x^2 - 2x - 7).
3x(5x2)+3x(−2x)+3x(−7)=15x3−6x2−21x3x(5x^2) + 3x(-2x) + 3x(-7) = 15x^3 - 6x^2 - 21x
The distributive property requires multiplying each term of the first polynomial by each term of the second polynomial.
2
Distribute the term −4-4 to every term in the trinomial (5x2−2x−7)(5x^2 - 2x - 7).
−4(5x2)−4(−2x)−4(−7)=−20x2+8x+28-4(5x^2) - 4(-2x) - 4(-7) = -20x^2 + 8x + 28
This completes the distribution for the second term of the binomial, taking care to properly multiply the negative signs.
3
Write out the fully expanded polynomial before combining like terms.
15x3−6x2−21x−20x2+8x+2815x^3 - 6x^2 - 21x - 20x^2 + 8x + 28
Listing all terms together makes it easier to identify which terms share the same variables and exponents.
4
Identify and combine the like terms that contain exactly the variable xx to the first power.
−21x+8x=−13x-21x + 8x = -13x
The question specifically asks for the coefficient of the xx term, so we combine −21x-21x and 8x8x.

Anahtar Kavram

Multiplying polynomials and combining like terms
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Multiplying Polynomials

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