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Difficulty: HardOperations on Polynomials

Two polynomial expressions are defined as P(x)=(2x−4)(x2−5x+c)P(x) = (2x - 4)(x^2 - 5x + c) and Q(x)=(x2+3x−4)(2x+a)Q(x) = (x^2 + 3x - 4)(2x + a), where aa and cc are constants. When P(x)P(x) is expanded and simplified, it has no xx term. If the coefficient of the x2x^2 term in the expanded and simplified form of Q(x)Q(x) is equal to the coefficient of the x2x^2 term in the expanded and simplified form of P(x)P(x), what is the value of aa?

Answer: -20

Answer

The value of aa is −20-20.
Expanding P(x)=(2x−4)(x2−5x+c)P(x) = (2x - 4)(x^2 - 5x + c) yields 2x3−14x2+(2c+20)x−4c2x^3 - 14x^2 + (2c + 20)x - 4c. Since there is no xx term, 2c+20=02c + 20 = 0, which means c=−10c = -10. This leaves the coefficient of the x2x^2 term in P(x)P(x) as −14-14. Expanding Q(x)=(x2+3x−4)(2x+a)Q(x) = (x^2 + 3x - 4)(2x + a) yields 2x3+(a+6)x2+(3a−8)x−4a2x^3 + (a + 6)x^2 + (3a - 8)x - 4a, so the coefficient of its x2x^2 term is a+6a + 6. Setting a+6=−14a + 6 = -14 and solving for aa gives a=−20a = -20.

Step-by-Step Solution

1
Expand the polynomial P(x)=(2x−4)(x2−5x+c)P(x) = (2x - 4)(x^2 - 5x + c) using the distributive property.
P(x)=2x3−14x2+(2c+20)x−4cP(x) = 2x^3 - 14x^2 + (2c + 20)x - 4c
To identify the coefficients of each term in P(x)P(x).
2
Set the coefficient of the xx term, 2c+202c + 20, to 00 and solve for cc.
c=−10c = -10
The problem states that P(x)P(x) has no xx term when simplified, which means its coefficient must be zero.
3
Determine the coefficient of the x2x^2 term in P(x)P(x).
The coefficient of x2x^2 is −14-14.
This coefficient will be equated to the x2x^2 coefficient of Q(x)Q(x) as per the problem constraints.
4
Expand the polynomial Q(x)=(x2+3x−4)(2x+a)Q(x) = (x^2 + 3x - 4)(2x + a) using the distributive property.
Q(x)=2x3+(a+6)x2+(3a−8)x−4aQ(x) = 2x^3 + (a + 6)x^2 + (3a - 8)x - 4a
To identify the coefficient of the x2x^2 term in Q(x)Q(x).
5
Set the coefficient of the x2x^2 term in Q(x)Q(x), which is a+6a + 6, equal to the coefficient of the x2x^2 term in P(x)P(x), which is −14-14, and solve for aa.
a=−20a = -20
The problem states that these two coefficients are equal.

Key Concept

Operations on Polynomials
Estimated Time:2m 30s
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Topics

Operations on Polynomials

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