Maths ch2 polynomials
37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.
Quick notes
The degree of a polynomial is the highest power of the variable. Degree 1 = linear, 2 = quadratic, 3 = cubic.
A zero of p(x) is a value k with p(k) = 0. Geometrically, the zeroes are the x-coordinates of the points where the graph of y = p(x) meets the x-axis.
The graph of a quadratic y = ax² + bx + c is a parabola. It opens upward if a > 0 and downward if a < 0. A quadratic has at most 2 zeroes; a cubic has at most 3.
If α and β are the zeroes of ax² + bx + c: α + β = −b/a and αβ = c/a.
A quadratic with sum of zeroes S and product P is k(x² − Sx + P), where k is any non-zero constant.
Part A: Multiple choice 16 questions
1. The degree of 5x³ − 2x² + 7 is
- 2
- 3
- 5
- 7
The highest power of x is 3.
2. The zero of p(x) = 2x + 3 is
- 3/2
- −2/3
- −3/2
- 2/3
2x + 3 = 0 gives x = −3/2.
3. The maximum number of zeroes a quadratic polynomial can have is
- 1
- 3
- 0
- 2
A polynomial of degree n has at most n zeroes.
4. The graph of y = ax² + bx + c (a ≠ 0) is a
- parabola
- straight line
- circle
- zig-zag line
Every quadratic polynomial has a parabola as its graph.
5. The graph of y = p(x) cuts the x-axis at exactly 3 points. The number of zeroes of p(x) is
- 2
- 3
- 4
- cannot say
Each point where the graph meets the x-axis gives one zero.
6. The sum of the zeroes of x² − 5x + 6 is
- −5
- 6
- 5
- −6
Sum = −b/a = −(−5)/1 = 5.
7. The product of the zeroes of 3x² − x − 4 is
- 4/3
- 1/3
- −1/3
- −4/3
Product = c/a = −4/3.
8. The zeroes of x² − 2x − 8 are
- 4 and −2
- −4 and 2
- 8 and −1
- 2 and 4
x² − 2x − 8 = (x − 4)(x + 2).
9. A quadratic polynomial whose sum of zeroes is 1/4 and product is −1 is
- 4x² + x − 4
- 4x² − x − 4
- x² − 4x − 1
- 4x² − x + 4
x² − (1/4)x − 1, multiplied by 4, gives 4x² − x − 4.
10. If 2 is a zero of x² + 3x + k, then k is
- 10
- −2
- −10
- 6
4 + 6 + k = 0 gives k = −10.
11. If α and β are the zeroes of x² − 4x + 3, then α + β + αβ is
- 1
- 4
- 3
- 7
α + β = 4 and αβ = 3, so the total is 7.
12. The graph of y = x² − 4 meets the x-axis at
- (2, 0) and (−2, 0)
- (4, 0) only
- (0, −4) only
- (0, 2) and (0, −2)
x² − 4 = 0 gives x = ±2.
13. A linear polynomial ax + b (a ≠ 0) has
- no zero
- exactly one zero
- two zeroes
- infinitely many zeroes
Its graph is a straight line that crosses the x-axis exactly once, at x = −b/a.
14. If a > 0, the parabola y = ax² + bx + c opens
- downward
- to the left
- upward
- to the right
A positive leading coefficient means the parabola opens upward.
15. The zeroes of 4s² − 4s + 1 are
- 1 and 1
- −1/2 and 1/2
- 2 and 2
- 1/2 and 1/2
4s² − 4s + 1 = (2s − 1)², so both zeroes are 1/2.
16. A quadratic polynomial with zeroes 0 and √5 is
- x² − √5x
- x² + √5x
- x² − 5
- x² + 5x
Sum = √5, product = 0, so x² − √5x.
Part B: Fill in the blanks 8 blanks
Part C: Questions and answers 9 questions
23. Find the zeroes of x² − 2x − 8 and verify the relationship between the zeroes and the coefficients.
Sum = 4 + (−2) = 2 = −(−2)/1 = −b/a ✓
Product = 4 × (−2) = −8 = −8/1 = c/a ✓
24. Find the zeroes of 6x² − 7x − 3 and verify the relationship.
Zeroes: 3/2 and −1/3.
Sum = 3/2 − 1/3 = 7/6 = −b/a ✓. Product = −1/2 = −3/6 = c/a ✓
25. Find the zeroes of 4u² + 8u.
Sum = −2 = −8/4 ✓. Product = 0 = 0/4 ✓
26. Find the zeroes of t² − 15.
Sum = 0 = −0/1 ✓. Product = −15 = −15/1 ✓
27. Find a quadratic polynomial whose sum and product of zeroes are −1/4 and 1/4.
Multiplying by 4: 4x² + x + 1.
28. Find a quadratic polynomial whose zeroes are √2 and −√2.
Polynomial: x² − 2.
29. If α and β are the zeroes of x² − 5x + 6, find α² + β².
α² + β² = (α + β)² − 2αβ = 25 − 12 = 13.
30. The graph of y = p(x) touches the x-axis at one point and does not cross it anywhere else. How many zeroes does p(x) have there?
31. Why can a quadratic polynomial not have three zeroes?