Polynomials
Study polynomial degrees, zeroes, factorization, and the division algorithm.
1. Polynomials, Degree and Types
- A polynomial in x is an expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the powers of x are whole numbers and the coefficients are real numbers.
- The highest power of x in p(x) is the degree of the polynomial. Degree 1 → linear (ax + b), degree 2 → quadratic (ax² + bx + c), degree 3 → cubic (ax³ + bx² + cx + d).
- A constant polynomial has degree 0; the zero polynomial has no defined degree.
- The value of p(x) at x = k is p(k), obtained by substituting k for x.
p(x) = aₙxⁿ + … + a₁x + a₀, aₙ ≠ 0Watch it happen — Degree & Number of Zeroes
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A polynomial of degree n has at most n zeroes and its graph meets the x-axis at most n times.
Q1.Define the degree of a polynomial and give one example each of a linear, quadratic and cubic polynomial.
2 marks2. Zeroes of a Polynomial
- A real number k is a zero of p(x) if p(k) = 0. For a linear polynomial ax + b, the only zero is −b/a.
- A polynomial of degree n has at most n zeroes: a linear polynomial has 1, a quadratic has at most 2, and a cubic has at most 3.
- Zeroes are also called the roots of the equation p(x) = 0.
Zero of ax + b is x = −b/aQ1.Find a quadratic polynomial whose sum and product of zeroes are −3 and 2 respectively.
3 marks3. Geometrical Meaning of the Zeroes
- The zeroes of p(x) are exactly the x-coordinates of the points where the graph y = p(x) cuts the x-axis.
- A linear polynomial gives a straight line that cuts the x-axis at exactly one point.
- A quadratic gives a parabola: open upward if a > 0, downward if a < 0. It can cut the x-axis at two points (two distinct zeroes), touch it at one point (two equal zeroes) or not meet it at all (no real zero).
- For a cubic, the graph can cut the x-axis at up to three points.
Watch it happen — Quadratic Graph Explorer
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The zeroes of a polynomial are the x-coordinates where the graph cuts the x-axis.
Q1.The graph of a quadratic polynomial does not intersect the x-axis. How many zeroes does it have and why?
2 marks4. Relationship between Zeroes and Coefficients
- For a quadratic polynomial ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a.
- To form a quadratic when the sum and product of zeroes are known: p(x) = k[x² − (α + β)x + αβ].
- For a cubic ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a.
- Example: for x² − 7x + 10, sum = 7 and product = 10, so the zeroes are 2 and 5.
α + β = −b/a, αβ = c/aWatch it happen — Nature of Roots
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Quadratic formula: x = (−b ± √D) / 2a.
Q1.Find the zeroes of x² − 2x − 8 and verify the relationship between the zeroes and the coefficients.
3 marks5. Division Algorithm for Polynomials
- If p(x) and g(x) are polynomials with g(x) ≠ 0, then there exist polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).
- Long division steps: divide the highest degree term of the dividend by the highest degree term of the divisor, multiply, subtract, and repeat with the new dividend.
- If two zeroes of a polynomial are known, divide by the corresponding factor to get the remaining factor and hence the remaining zeroes.
- A useful check: g(x) is a factor of p(x) exactly when the remainder r(x) is 0.
p(x) = g(x)·q(x) + r(x), deg r < deg gQ1.Divide 3x³ + x² + 2x + 5 by x² + 2x + 1 and verify the division algorithm.
5 marks6. Board Exam Tips for Polynomials
- Always compare the given polynomial with ax² + bx + c carefully — a wrong sign in b changes the sum of zeroes.
- Verify your zeroes by substituting them back into p(x); the value must come out to 0.
- For graph-based questions, count the number of points where the curve meets the x-axis — that count is the number of zeroes.
- Factorisation by splitting the middle term is faster than the quadratic formula whenever the numbers are simple.
Watch it happen — Completing the Square
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Completing the square shows the vertex directly and derives the quadratic formula.