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Difficulty: Very hardOperations on Polynomials

When the product of the polynomials (x3−3x2+2x−4)(x^3 - 3x^2 + 2x - 4) and (ax2+bx+c)(ax^2 + bx + c) is subtracted from 2x5−5x4+5x3−18x2+4x−162x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16, the resulting polynomial is equal to 00 for all real values of xx. What is the value of 4a+2b−c4a + 2b - c?

Answer: 6

Answer

6
The correct answer is 6. By equating the corresponding coefficients of the product (x3−3x2+2x−4)(ax2+bx+c)(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) to the polynomial 2x5−5x4+5x3−18x2+4x−162x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16, we find a=2a = 2 from the x5x^5 terms, c=4c = 4 from the constant terms, and b=1b = 1 from the x4x^4 terms. Substituting these values into 4a+2b−c4a + 2b - c yields 4(2)+2(1)−4=64(2) + 2(1) - 4 = 6.

Step-by-Step Solution

1
Relate the product of the polynomials to the given polynomial expression.
(x3−3x2+2x−4)(ax2+bx+c)=2x5−5x4+5x3−18x2+4x−16(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16
Since subtracting the product from the given polynomial results in a polynomial that is always 00, the product must be identically equal to that polynomial.
2
Equate the leading coefficients to find the value of aa.
a=2a = 2
The highest-degree term of the product is x3⋅ax2=ax5x^3 \cdot ax^2 = ax^5, which must equal the highest-degree term on the right side, 2x52x^5.
3
Equate the constant terms to find the value of cc.
−4c=−16  ⟹  c=4-4c = -16 \implies c = 4
The constant term of the product is −4⋅c=−4c-4 \cdot c = -4c, which must equal the constant term on the right side, −16-16.
4
Equate the coefficients of the x4x^4 terms to find the value of bb.
b−3a=−5  ⟹  b−6=−5  ⟹  b=1b - 3a = -5 \implies b - 6 = -5 \implies b = 1
The x4x^4 term in the expanded product comes from (x3)(bx)+(−3x2)(ax2)=(b−3a)x4(x^3)(bx) + (-3x^2)(ax^2) = (b - 3a)x^4, which must equal the x4x^4 term on the right side, −5x4-5x^4.
5
Compute the value of the requested expression 4a+2b−c4a + 2b - c.
4(2)+2(1)−4=64(2) + 2(1) - 4 = 6
Substitute the determined values a=2a = 2, b=1b = 1, and c=4c = 4 into the expression.

Key Concept

Operations on polynomials, specifically multiplication, subtraction, and equating corresponding coefficients.

Alternative Method

Alternatively, evaluate the polynomial equation at x=2x = 2. Substituting x=2x = 2 into (x3−3x2+2x−4)(ax2+bx+c)=2x5−5x4+5x3−18x2+4x−16(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16 gives (8−12+4−4)(4a+2b+c)=64−80+40−72+8−16(8 - 12 + 4 - 4)(4a + 2b + c) = 64 - 80 + 40 - 72 + 8 - 16, which simplifies to −4(4a+2b+c)=−56-4(4a + 2b + c) = -56. Dividing both sides by −4-4 yields 4a+2b+c=144a + 2b + c = 14. Since equating the constant terms gives −4c=−16  ⟹  c=4-4c = -16 \implies c = 4, we can substitute c=4c = 4 into 4a+2b+c=144a + 2b + c = 14 to get 4a+2b+4=14  ⟹  4a+2b=104a + 2b + 4 = 14 \implies 4a + 2b = 10. Subtracting c=4c=4 from both sides gives the desired expression value: 4a+2b−c=10−4=64a + 2b - c = 10 - 4 = 6.
Estimated Time:2m 30s
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Topics

Operations on Polynomials

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