Polynomials
This chapter covers polynomials and their degree, zeroes of a polynomial, the geometric meaning of zeroes, and the relationship between zeroes and coefficients for quadratic and cubic polynomials.
A polynomial in x is an expression of the form anxn + an-1xn-1 + ... + a1x + a0, where an ≠ 0. The highest power of x is called the degree. Degree 1 = linear, degree 2 = quadratic, degree 3 = cubic.
A real number k is called a zero of polynomial p(x) if p(k) = 0.
The zeroes of p(x) are exactly the x-coordinates of the points where the graph y = p(x) intersects the x-axis. A linear polynomial has exactly 1 zero, a quadratic has at most 2, a cubic has at most 3 - in general, a degree-n polynomial has at most n zeroes.
For a quadratic polynomial ax2 + bx + c (a ≠ 0) with zeroes α and β: sum of zeroes α+β = -b/a, and product of zeroes αβ = c/a. For a cubic polynomial ax3+bx2+cx+d with zeroes α, β, γ: α+β+γ = -b/a, αβ+βγ+γα = c/a, and αβγ = -d/a.
Find the zeroes of the quadratic polynomial x² - 2x - 8, and verify the relationship between the zeroes and the coefficients.
x² - 2x - 8 = x² - 4x + 2x - 8 = x(x-4) + 2(x-4) = (x-4)(x+2). Zeroes are 4 and -2. Sum = 4 + (-2) = 2 = -(-2)/1 = -b/a. Product = 4 × (-2) = -8 = -8/1 = c/a. Verified.
Find the zeroes of the quadratic polynomial 4s² - 4s + 1, and verify the relationship between the zeroes and the coefficients.
4s² - 4s + 1 = (2s-1)². Zeroes are 1/2 and 1/2 (repeated). Sum = 1/2+1/2 = 1 = -(-4)/4 = -b/a. Product = 1/4 = 1/4 = c/a. Verified.
Find the zeroes of the quadratic polynomial 6x² - 3 - 7x, and verify the relationship between the zeroes and the coefficients.
Rewrite as 6x² - 7x - 3. 6x² - 9x + 2x - 3 = 3x(2x-3) + 1(2x-3) = (2x-3)(3x+1). Zeroes are 3/2 and -1/3. Sum = 3/2 - 1/3 = 7/6 = -(-7)/6 = -b/a. Product = (3/2)(-1/3) = -1/2 = -3/6 = c/a. Verified.
Find the zeroes of the quadratic polynomial 4u² + 8u, and verify the relationship between the zeroes and the coefficients.
4u² + 8u = 4u(u+2). Zeroes are 0 and -2. Sum = -2 = -8/4 = -b/a. Product = 0 = 0/4 = c/a. Verified.
Find the zeroes of the quadratic polynomial t² - 15, and verify the relationship between the zeroes and the coefficients.
t² - 15 = (t-√15)(t+√15). Zeroes are √15 and -√15. Sum = 0 = -0/1 = -b/a. Product = -15 = -15/1 = c/a. Verified.
Find the zeroes of the quadratic polynomial 3x² - x - 4, and verify the relationship between the zeroes and the coefficients.
3x² - x - 4 = 3x² - 4x + 3x - 4 = x(3x-4) + 1(3x-4) = (3x-4)(x+1). Zeroes are 4/3 and -1. Sum = 4/3 - 1 = 1/3 = -(-1)/3 = -b/a. Product = (4/3)(-1) = -4/3 = -4/3 = c/a. Verified.
Find a quadratic polynomial whose sum and product of zeroes are 1/4 and -1 respectively.
A quadratic with sum S and product P of zeroes is k[x² - Sx + P]. Here: x² - (1/4)x + (-1) = x² - x/4 - 1, or multiplying by 4: 4x² - x - 4.
Find a quadratic polynomial whose sum and product of zeroes are √2 and 1/3 respectively.
x² - √2x + 1/3, or multiplying by 3: 3x² - 3√2x + 1.
Find a quadratic polynomial whose sum and product of zeroes are 0 and √5 respectively.
x² - 0x + √5 = x² + √5.
Find a quadratic polynomial whose sum and product of zeroes are 1 and 1 respectively.
x² - x + 1.
Find a quadratic polynomial whose sum and product of zeroes are -1/4 and 1/4 respectively.
x² + x/4 + 1/4, or multiplying by 4: 4x² + x + 1.
Find a quadratic polynomial whose sum and product of zeroes are 4 and 1 respectively.
x² - 4x + 1.
Verify that 3, -1, -1/3 are the zeroes of the cubic polynomial p(x) = 3x³ - 5x² - 11x - 3, and verify the relationship between its zeroes and coefficients.
p(3) = 3(27) - 5(9) - 11(3) - 3 = 81 - 45 - 33 - 3 = 0. p(-1) = 3(-1) - 5(1) - 11(-1) - 3 = -3 - 5 + 11 - 3 = 0. p(-1/3) = 3(-1/27) - 5(1/9) - 11(-1/3) - 3 = -1/9 - 5/9 + 11/3 - 3 = -6/9 + 33/9 - 27/9 = 0. All verified. Here a=3, b=-5, c=-11, d=-3. Sum = 3 + (-1) + (-1/3) = 5/3 = -(-5)/3 = -b/a. Sum of products in pairs = (3)(-1)+(-1)(-1/3)+(-1/3)(3) = -3+1/3-1 = -11/3 = c/a. Product = (3)(-1)(-1/3) = 1, but -d/a = -(-3)/3 = 1. Verified.
Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time, and product of its zeroes as 2, -7, and -14 respectively.
A cubic with these relations is x³ - (sum)x² + (sum of products)x - (product) = x³ - 2x² - 7x + 14.
If one of the zeroes of the quadratic polynomial x² + 3x + k is 2, the value of k is:
Correct answer: (D) -10. Since 2 is a zero: 4 + 6 + k = 0, so k = -10.
The graph of y = p(x) for a quadratic polynomial p(x) can intersect the x-axis at most at how many points?
Correct answer: (C) 2.
A quadratic polynomial can have at most 2 zeroes.
True - the number of zeroes of a polynomial can never exceed its degree.
For a quadratic polynomial ax² + bx + c with zeroes α and β, the product αβ equals _______.
c/a
The number of zeroes of a polynomial p(x) is at most equal to its degree. A linear polynomial has exactly one zero; a non-zero constant polynomial has no zero.