1. Polynomial and its degree

A polynomial in x is an expression such as 3x² − 5x + 2. Its degree is the highest power of x with a non-zero coefficient. A quadratic polynomial has degree 2.

Important forms

• Linear: ax + b
• Quadratic: ax² + bx + c, a ≠ 0
• Cubic: ax³ + bx² + cx + d

2. Geometrical meaning of zeroes

A zero of p(x) is a value k for which p(k)=0. Graphically, the zeroes of a polynomial are the x-coordinates where its graph meets the x-axis.

3. Relationship between zeroes and coefficients

For p(x)=ax²+bx+c with zeroes α and β:

4. Finding a polynomial from its zeroes

If α and β are known, a quadratic polynomial with these zeroes is k[x²−(α+β)x+αβ], where k is any non-zero constant.

Example: Zeroes 2 and −5 → x²+3x−10.

5. Division algorithm for polynomials

For polynomials p(x) and a non-zero polynomial g(x), there exist unique q(x) and r(x) such that:

where either r(x)=0 or degree r(x) < degree g(x).

Example: Dividing x²+5x+6 by x+2 gives quotient x+3 and remainder 0.

6. Common mistakes

• Confusing zero of a polynomial with its coefficient.
• Forgetting the negative sign in α+β=−b/a.
• Using c/a as the sum instead of the product.
• Ignoring the degree condition in polynomial division.

7. Competency Case

A rectangular garden has a side represented by x metres and its area is described by x²−7x+12. The expression factors as (x−3)(x−4).

1. Identify the possible dimensions.
2. What are the zeroes?
3. Verify the sum and product of the zeroes using coefficients.

8. Practice Set

1. Find the zeroes of 3x²−8x+4.
2. If the zeroes of x²−px+q are 2 and 5, find p and q.
3. Form a quadratic polynomial whose zeroes are −3 and 7.
4. Divide 2x³+3x²−11x−6 by x+2.
5. If one zero of 2x²+kx+3 is 1, find k and the other zero.

9. Quick Revision

Zero → x-intercept → α+β=−b/a → αβ=c/a → construct polynomial → polynomial division.