Polynomials

Degree of a polynomial

The highest power of the variable in the polynomial. For instance 4x + 2 has degree 1, 2y² − 3y + 4 has degree 2 and 5x³ − 4x² + x − 2 has degree 3; polynomials of degree one, two and three are named linear, quadratic and cubic.

Zero of a linear polynomial ax + b

−b/a, that is −(constant term) ÷ (coefficient of x), where a ≠ 0. For 2x + 3 the zero is −3/2, the point where the line y = 2x + 3 crosses the x-axis.

Quadratic polynomial

A polynomial of degree 2, whose general form in x is ax² + bx + c with a, b and c real and a ≠ 0. The name comes from 'quadrate', meaning square.

Cubic polynomial

A polynomial of degree 3, whose most general form is ax³ + bx² + cx + d with a, b, c and d real and a ≠ 0. Examples include x³, 2 − x³ and 3x³ − 2x² + x − 1.

Zero of a polynomial

A real number k for which p(k) = 0. For p(x) = x² − 3x − 4, both p(−1) and p(4) equal 0, so −1 and 4 are its zeroes.

Geometrical meaning of zeroes

The zeroes of p(x) are exactly where the graph of y = p(x) crosses or touches the x-axis: each zero is the x-coordinate of such a point. So −1 and 4, the zeroes of x² − 3x − 4, are where its parabola cuts the x-axis.

In how many ways can the graph of a quadratic polynomial meet the x-axis, and what does each mean for its zeroes?

The graph is a parabola, opening upwards if a > 0 and downwards if a < 0. It can cut the x-axis at two distinct points, giving two zeroes; touch it at one point, giving two equal zeroes; or miss it entirely, giving no zero.

Maximum number of zeroes

A polynomial of degree n has at most n zeroes, because its graph meets the x-axis in at most n points. So a quadratic has at most 2 zeroes and a cubic at most 3.

How do you find the number of zeroes of p(x) from the graph of y = p(x)?

Count the points where the graph meets the x-axis; the x-coordinate of each is a zero. A graph meeting the x-axis at one point gives one zero, at two points two zeroes, and so on.

Sum of zeroes of a quadratic

α + β = −b/a = −(coefficient of x) ÷ (coefficient of x²), for zeroes α and β of ax² + bx + c. For 2x² − 8x + 6, whose zeroes are 1 and 3, the sum 4 equals −(−8)/2.

Product of zeroes of a quadratic

αβ = c/a = (constant term) ÷ (coefficient of x²), for zeroes α and β of ax² + bx + c. For 2x² − 8x + 6, whose zeroes are 1 and 3, the product 3 equals 6/2.

Why are the sum and product of the zeroes of ax² + bx + c equal to −b/a and c/a?

If α and β are zeroes, then (x − α) and (x − β) are factors, so ax² + bx + c = k[x² − (α + β)x + αβ]. Comparing coefficients gives a = k, b = −k(α + β) and c = kαβ, which rearrange to α + β = −b/a and αβ = c/a.

Quadratic polynomial with given zeroes α and β

k[x² − (α + β)x + αβ] for a non-zero real constant k. In words: x² − (sum of zeroes)x + (product of zeroes), multiplied by any such constant.

How do you find the zeroes of a quadratic by factorising and verify the zeroes-coefficients relationship?

Factorise, set each factor equal to zero, then compare the sum and product of the zeroes with −b/a and c/a. In the chapter's example, x² + 7x + 10 = (x + 2)(x + 5) gives zeroes −2 and −5, whose sum −7 equals −7/1 and whose product 10 equals 10/1.

How do you find a quadratic polynomial when the sum and product of its zeroes are given?

Take a = 1, so that b = −(sum) and c = product, and write x² − (sum)x + product. In the chapter's example, a sum of −3 and a product of 2 give x² + 3x + 2; every other answer has the form k(x² + 3x + 2).

Relationships between zeroes and coefficients of a cubic polynomial

If α, β and γ are the zeroes of ax³ + bx² + cx + d, then α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a.

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