Polynomials — Practice

Question 1

If α and β are zeroes of p(x) = 3x² − 5x + 2, find (a) α + β, (b) αβ, (c) α² + β².

Accepted Answer

Step 1 — Identify a = 3, b = −5, c = 2.
  α + β = −b/a = −(−5)/3 = 5/3.
  αβ = c/a = 2/3.

Step 2 — Compute α² + β² using identity.
  α² + β² = (α + β)² − 2αβ
           = (5/3)² − 2(2/3)
           = 25/9 − 4/3
           = 25/9 − 12/9 = 13/9.

✦ Answer: α + β = 5/3, αβ = 2/3, α² + β² = 13/9.

Question 2

Find a quadratic polynomial whose zeroes are 3 and −5.

Accepted Answer

Step 1 — Compute sum and product.
  Sum = 3 + (−5) = −2.
  Product = 3 × (−5) = −15.

Step 2 — Form the polynomial.
  p(x) = x² − (sum)x + (product) = x² − (−2)x + (−15) = x² + 2x − 15.

Step 3 — Verify by factorising.
  x² + 2x − 15 = (x + 5)(x − 3). Zeroes: x = −5 and x = 3. ✓

✦ Answer: x² + 2x − 15.

Question 3

Verify that 1, 2, 3 are zeroes of p(x) = x³ − 6x² + 11x − 6. Also verify the relationship between zeroes and coefficients.

Accepted Answer

Step 1 — Verify by substitution.
  p(1) = 1 − 6 + 11 − 6 = 0 ✓
  p(2) = 8 − 24 + 22 − 6 = 0 ✓
  p(3) = 27 − 54 + 33 − 6 = 0 ✓

Step 2 — Identify a = 1, b = −6, c = 11, d = −6.

Step 3 — Verify sum of zeroes.
  α + β + γ = 1 + 2 + 3 = 6. −b/a = −(−6)/1 = 6. ✓

Step 4 — Verify pairwise products.
  αβ + βγ + γα = 1×2 + 2×3 + 3×1 = 2 + 6 + 3 = 11. c/a = 11/1 = 11. ✓

Step 5 — Verify triple product.
  αβγ = 1×2×3 = 6. −d/a = −(−6)/1 = 6. ✓

✦ Answer: All zeroes verified; all three coefficient relationships confirmed.

Question 4

Divide p(x) = x³ − 3x² + 5x − 3 by g(x) = x² − 2. Find the quotient q(x) and remainder r(x). Verify.

Accepted Answer

Step 1 — Set up long division.
  x³ − 3x² + 5x − 3 divided by x² − 2.

Step 2 — First term of quotient.
  x³ ÷ x² = x. Multiply: x(x² − 2) = x³ − 2x.
  Subtract: (x³ − 3x² + 5x − 3) − (x³ − 2x) = −3x² + 7x − 3.

Step 3 — Second term of quotient.
  −3x² ÷ x² = −3. Multiply: −3(x² − 2) = −3x² + 6.
  Subtract: (−3x² + 7x − 3) − (−3x² + 6) = 7x − 9.

Step 4 — Degree of 7x − 9 (degree 1) < degree of x² − 2 (degree 2). Stop.
  q(x) = x − 3, r(x) = 7x − 9.

Step 5 — Verify: g(x) · q(x) + r(x) = (x² − 2)(x − 3) + (7x − 9)
  = x³ − 3x² − 2x + 6 + 7x − 9 = x³ − 3x² + 5x − 3 = p(x). ✓

✦ Answer: Quotient = x − 3, Remainder = 7x − 9.

Question 5

If two zeroes of p(x) = x³ + 3x² − 5x − 15 are √5 and −√5, find the third zero.

Accepted Answer

Step 1 — Use the sum of zeroes.
  Let the three zeroes be √5, −√5, and γ.
  Sum = √5 + (−√5) + γ = γ = −b/a = −3/1 = −3.

Step 2 — Verify with product.
  Triple product = √5 × (−√5) × (−3) = (−5)(−3) = 15.
  −d/a = −(−15)/1 = 15. ✓

✦ Answer: Third zero = −3.

Question 6

On dividing x³ − 3x² + x + 2 by g(x), the quotient is x − 2 and remainder is −2x + 4. Find g(x).

Accepted Answer

Step 1 — Use the division algorithm.
  p(x) = g(x) · q(x) + r(x)
  x³ − 3x² + x + 2 = g(x) · (x − 2) + (−2x + 4).

Step 2 — Rearrange for g(x).
  g(x) · (x − 2) = p(x) − r(x)
  = (x³ − 3x² + x + 2) − (−2x + 4)
  = x³ − 3x² + 3x − 2.

Step 3 — Divide by (x − 2) to get g(x).
  x³ − 3x² + 3x − 2 ÷ (x − 2):
  x³ ÷ x = x². x²(x−2) = x³ − 2x². Subtract: −x² + 3x − 2.
  −x² ÷ x = −x. −x(x−2) = −x² + 2x. Subtract: x − 2.
  x ÷ x = 1. 1(x−2) = x − 2. Subtract: 0.
  So g(x) = x² − x + 1.

✦ Answer: g(x) = x² − x + 1.

Practice — Polynomials

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