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Polynomials in ℂ

16 practice questions with full worked solutions. Free, no account needed.

Problems & worked solutions

Problem #0441 International

Problem 1 — Polynomials in ℂ

The degree of the polynomial P(X)=2X4−5X3+X2−7 is:

Show answer & worked solution
  1. A. 2
  2. B. 3
  3. C. 4✓ correct
  4. D. 7

The degree is the largest exponent of X with a non-zero coefficient. Here it is 4.

Problem #0442 International

Problem 2 — Polynomials in ℂ

For P(X)=X3−2X+5, the value of P(2) is:

Show answer & worked solution
  1. A. 5
  2. B. 7
  3. C. 9✓ correct
  4. D. 13

P(2)=8−4+5=9.

Problem #0443 International

Problem 3 — Polynomials in ℂ

For A=X2+3X−1 and B=2X2−X+4, the polynomial A+B equals:

Show answer & worked solution
  1. A. X2+2X+3
  2. B. 3X2+4X+5
  3. C. 3X2+2X+3✓ correct
  4. D. −X2+4X−5

Add coefficients: (1+2)X2+(3−1)X+(−1+4)=3X2+2X+3.

Problem #0444 International

Problem 4 — Polynomials in ℂ

The remainder of the division of P(X)=X3+2X2−X+5 by X−1 is:

Show answer & worked solution
  1. A. 0
  2. B. 5
  3. C. 7✓ correct
  4. D. 9

P(1)=1+2−1+5=7.

Problem #0448 International

Problem 5 — Polynomials in ℂ

The polynomial P(X)=X3−X2−4X+4 factors as:

Show answer & worked solution
  1. A. (X−1)(X−2)(X+2)✓ correct
  2. B. (X+1)(X−2)(X+2)
  3. C. (X−1)2(X+4)
  4. D. (X−1)(X+2)2

P(1)=0, so X−1 divides P. By Horner: P(X)=(X−1)(X2−4)=(X−1)(X−2)(X+2).

Problem #0445 International

Problem 6 — Polynomials in ℂ

The quotient when P(X)=X3−4X2+5X−2 is divided by X−2 has degree:

Show answer & worked solution
  1. A. 1
  2. B. 2✓ correct
  3. C. 3
  4. D. 4

deg⁡(Q)=3−1=2.

Problem #0447 International

Problem 7 — Polynomials in ℂ

For the polynomial P(X)=2X3−6X2+X−4 with roots x1,x2,x3, the product x1x2x3 equals:

Show answer & worked solution
  1. A. −4
  2. B. −2
  3. C. 2✓ correct
  4. D. 4

x1x2x3=−−42=2.

Problem #0446 International

Problem 8 — Polynomials in ℂ

For P(X)=X3−6X2+11X−6 with roots x1,x2,x3, the value of x1+x2+x3 is:

Show answer & worked solution
  1. A. −11
  2. B. −6
  3. C. 6✓ correct
  4. D. 11

The leading coefficient is 1 and the coefficient of X2 is −6, so x1+x2+x3=−(−6)=6. (The roots are in fact 1,2,3.)

Problem #0916 US SAT

Problem 9 — Division, Bezout, Horner

Dividing x2−5x+6 by (x−2) gives the quotient:

Show answer & worked solution
  1. A. x−6
  2. B. x+3
  3. C. x−3✓ correct
  4. D. x−2

x2−5x+6=(x−2)(x−3).

So x2−5x+6x−2=x−3, with no remainder.

Problem #0914 US SAT

Problem 10 — Factorization in C and R

Factor over R: x2−9.

Show answer & worked solution
  1. A. (x−3)(x+3)✓ correct
  2. B. (x−3)2
  3. C. (x+3)2
  4. D. (x−9)(x+1)

x2−9=x2−32=(x−3)(x+3) — the difference-of-squares identity.

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