A polynomial in one variable x is an expression of the form an · xn + an-1 · xn-1 + ... + a1 · x + a0, where the coefficients are real numbers and n is a non-negative integer. Polynomials are fundamental to algebra, and understanding their zeros and factors is critical for higher mathematics.
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Key Concepts
Degree of a Polynomial: The highest power of the variable. Example: 3x3 - 2x + 1 has degree 3 (cubic).
- Types by Degree:
- Linear: degree 1 (e.g. 2x + 3)
- Quadratic: degree 2 (e.g. x2 - 5x + 6)
- Cubic: degree 3 (e.g. x3 - 2x2 + x - 1)
Zero (Root) of a Polynomial: A value k such that p(k) = 0. A polynomial of degree n has at most n zeros.
- Geometrical Meaning: The zeros of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
- A linear polynomial has exactly 1 zero (graph is a straight line crossing x-axis once).
- A quadratic polynomial can have 0, 1, or 2 zeros (graph is a parabola).
- A cubic polynomial has 1 or 3 real zeros.
- Relationship between Zeros and Coefficients (Quadratic):
- For p(x) = ax2 + bx + c with zeros alpha and beta:
- Sum of zeros: alpha + beta = -b/a
- Product of zeros: alpha x beta = c/a
- Relationship between Zeros and Coefficients (Cubic):
- For p(x) = ax3 + bx2 + cx + d with zeros alpha, beta, gamma:
- alpha + beta + gamma = -b/a
- alpha · beta + beta · gamma + gamma · alpha = c/a
- alpha · beta · gamma = -d/a
Division Algorithm for Polynomials:
If p(x) and g(x) are polynomials (g(x) not 0), then:
p(x) = g(x) x q(x) + r(x), where degree of r(x) < degree of g(x) or r(x) = 0.
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Worked Examples
Find the zeros of p(x) = x2 - 5x + 6.
Factorise: x2 - 5x + 6 = (x - 2)(x - 3).
Zeros: x = 2 and x = 3.
Check: Sum = 2 + 3 = 5 = -(-5)/1 = b/a. Product = 2 x 3 = 6 = 6/1 = c/a. Correct!
Find a quadratic polynomial whose zeros are 3 and -4.
Sum = 3 + (-4) = -1; Product = 3 x (-4) = -12.
Polynomial: x2 - (sum)x + product = x2 + x - 12.
If the zeros of x2 + 7x + 10 are alpha and beta, find alpha2 + beta2.
alpha + beta = -7, alpha · beta = 10.
alpha2 + beta2 = (alpha + beta)2 - 2 · alpha · beta = (-7)2 - 2(10) = 49 - 20 = 29.
Verify that 2, -1, and -3 are zeros of x3 + 2x2 - 5x - 6.
p(2) = 8 + 8 - 10 - 6 = 0. p(-1) = -1 + 2 + 5 - 6 = 0. p(-3) = -27 + 18 + 15 - 6 = 0. All verified!
Divide p(x) = 2x3 + x2 - 5x + 2 by g(x) = 2x - 1.
Using long division: quotient = x2 + x - 2, remainder = 0.
So 2x3 + x2 - 5x + 2 = (2x - 1)(x2 + x - 2) = (2x-1)(x+2)(x-1).
Zeros: x = 1/2, x = -2, x = 1.
A quadratic polynomial has sum of zeros = 0 and product of zeros = -16. Write the polynomial.
p(x) = x2 - 0 · x + (-16) = x2 - 16.
If one zero of 3x2 - 8x + p is the reciprocal of the other, find p.
Let zeros be alpha and 1/alpha. Product = alpha x (1/alpha) = 1 = p/3. So p = 3.
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Key Formulas
- Sum of zeros (quadratic): alpha + beta = -b/a
- Product of zeros (quadratic): alpha · beta = c/a
- General quadratic from zeros: x2 - (alpha+beta)x + alpha · beta
- Division: p(x) = g(x) · q(x) + r(x)
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Common mistakes
- Mixing up the sign: sum of zeros = -b/a, not b/a.
- Forgetting that a polynomial of degree n has AT MOST n zeros (could have fewer real zeros).
- Confusing degree 0 (non-zero constant) with the zero polynomial (undefined degree).
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Summary
Polynomials chapter builds the bridge between algebraic expressions and their graphs. The key skill is using zero-coefficient relationships to find unknown values without actually solving the polynomial each time.