Chapter 2: Polynomials
Polynomials and their zeroes
The highest power of x in a polynomial p(x) is its degree. Degree 1: linear (ax + b). Degree 2: quadratic (ax² + bx + c, a ≠ 0). Degree 3: cubic. The value got by putting x = k in p(x) is p(k). If p(k) = 0, then k is a zero of the polynomial. The zero of ax + b is −b/a. For example, the zeroes of x² − 3x − 4 are −1 and 4.
Geometrical meaning of zeroes
The x-coordinates of the points where the graph of y = p(x) meets the x-axis are the zeroes. The graph of a linear polynomial is a straight line with exactly one zero. The graph of a quadratic is a parabola (opening up if a > 0, down if a < 0), and it meets the x-axis at two points, one point or no point, that is 2, 1 or 0 zeroes. A polynomial of degree n has at most n zeroes. (Drawing the graphs yourself is not asked in the examination.)
Relationship between zeroes and coefficients
If α and β are the zeroes of the quadratic ax² + bx + c, then α + β = −b/a and αβ = c/a. Example: 2x² − 8x + 6 = 2(x − 1)(x − 3), zeroes 1 and 3. Sum 4 = 8/2 and product 3 = 6/2.
Forming a polynomial from sum and product
If the sum of the zeroes is S and the product is P, one quadratic polynomial is x² − Sx + P. Any other such polynomial has the form k(x² − Sx + P). For sum −3 and product 2 we get x² + 3x + 2. (The relations among the zeroes of a cubic polynomial are extra information, not for the examination.)
Important questions
Q1. What is the degree of a polynomial? Write the general forms of linear, quadratic and cubic polynomials.
The highest power of x in the polynomial is its degree. Linear: ax + b (a ≠ 0). Quadratic: ax² + bx + c (a ≠ 0). Cubic: ax³ + bx² + cx + d (a ≠ 0).
Q2. Find the value of p(x) = x² − 3x − 4 at x = 2.
p(2) = 2² − 3 × 2 − 4 = 4 − 6 − 4 = −6.
Q3. What is a zero of a polynomial?
A real number k for which p(k) = 0. For example, the zeroes of x² − 3x − 4 are −1 and 4, because p(−1) = 0 and p(4) = 0.
Q4. What is the zero of the linear polynomial ax + b? Find the zero of 2x + 3.
The zero of ax + b is −b/a. So the zero of 2x + 3 is −3/2.
Q5. What is the geometrical meaning of a zero?
The x-coordinates of the points where the graph of y = p(x) meets the x-axis are the zeroes of p(x).
Q6. What does the graph of a quadratic polynomial look like?
A parabola. It opens upwards if a > 0 and downwards if a < 0.
Q7. State the three possible ways the graph of a quadratic polynomial can meet the x-axis.
It can (i) cut the x-axis at two distinct points (two zeroes), (ii) touch it at exactly one point (one zero, two equal), or (iii) not meet it at all (no zero).
Q8. What is the maximum number of zeroes a polynomial of degree n can have?
At most n zeroes. So a quadratic has at most 2 zeroes and a cubic has at most 3.
Q9. Write the formulae for the sum and product of the zeroes α and β of ax² + bx + c.
α + β = −b/a and αβ = c/a.
Q10. Find the zeroes of x² − 2x − 8 and verify the relationship.
x² − 2x − 8 = (x − 4)(x + 2), zeroes 4 and −2. Sum = 2 = −(−2)/1 and product = −8 = −8/1, so the relationship holds.
Q11. Find the zeroes of 4s² − 4s + 1 and verify the relationship.
4s² − 4s + 1 = (2s − 1)², zeroes 1/2 and 1/2. Sum = 1 = 4/4 and product = 1/4, so the relationship holds.
Q12. Find the zeroes of 6x² − 3 − 7x and verify the relationship.
Write it as 6x² − 7x − 3 = (3x + 1)(2x − 3). Zeroes −1/3 and 3/2. Sum = 7/6 = −(−7)/6 and product = −1/2 = −3/6, so the relationship holds.
Q13. Find the zeroes of 4u² + 8u and verify the relationship.
4u² + 8u = 4u(u + 2), zeroes 0 and −2. Sum = −2 = −8/4 and product = 0 = 0/4, so the relationship holds.
Q14. Find the zeroes of t² − 15 and verify the relationship.
t² − 15 = (t − √15)(t + √15), zeroes √15 and −√15. Sum = 0 = −0/1 and product = −15 = −15/1, so the relationship holds.
Q15. Find the zeroes of 3x² − x − 4 and verify the relationship.
3x² − x − 4 = (3x − 4)(x + 1), zeroes 4/3 and −1. Sum = 1/3 = −(−1)/3 and product = −4/3, so the relationship holds.
Q16. Find a quadratic polynomial whose zeroes have sum 1/4 and product −1.
x² − (sum)x + (product) = x² − x/4 − 1. Multiplying by 4 gives 4x² − x − 4. (Any k(4x² − x − 4) also works.)
Q17. Find a quadratic polynomial whose zeroes have sum √2 and product 1/3.
x² − √2 x + 1/3. Multiplying by 3 gives 3x² − 3√2 x + 1.
Q18. Find a quadratic polynomial whose zeroes have sum 0 and product √5.
x² − 0·x + √5 = x² + √5.
Q19. Find a quadratic polynomial whose zeroes have sum −3 and product 2.
x² − (−3)x + 2 = x² + 3x + 2. Any other such polynomial will be of the form k(x² + 3x + 2).
Q20. Find the zeroes of x² + 7x + 10 and verify the relationship.
x² + 7x + 10 = (x + 2)(x + 5), zeroes −2 and −5. Sum = −7 = −7/1 and product = 10 = 10/1, so the relationship holds.
Quiz
1. What is the zero of 2x + 3?
The zero of ax + b is −b/a, that is −3/2.
2. What is the maximum number of zeroes a quadratic polynomial can have?
A polynomial of degree 2 has at most 2 zeroes.
3. What is the sum of the zeroes of x² − 2x − 8?
Sum = −b/a = −(−2)/1 = 2.
4. What is the product of the zeroes of 3x² − x − 4?
Product = c/a = −4/3.
5. A parabola lies completely above the x-axis. How many zeroes does the quadratic polynomial have?
The graph does not meet the x-axis, so there is no zero.