Definition of Polynomials

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. It can be expressed in the form:

\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]

where \( a_0, a_1, \ldots, a_n \) are real numbers and \( n \) is a whole number called the degree of the polynomial.

Types of polynomials based on number of terms:

• Monomial: A polynomial with one term, e.g., \( 5x^3 \).
• Binomial: A polynomial with two terms, e.g., \( x^2 + 3x \).
• Trinomial: A polynomial with three terms, e.g., \( x^2 + 5x + 6 \).

Worked Example

Identify whether the following are polynomials and classify them:

• \( 4x^3 - 2x + 7 \)
• \( 3x^{-2} + 5 \)
• \( 2x^2 + 3x + 1 \)

Solution:

• \( 4x^3 - 2x + 7 \) is a polynomial of degree 3 with three terms (trinomial).
• \( 3x^{-2} + 5 \) is not a polynomial because of the negative exponent.
• \( 2x^2 + 3x + 1 \) is a polynomial of degree 2 with three terms (trinomial).

Practice Set

• Level 1: Identify if \( 7x^4 + 3x^2 - 5 \) is a polynomial.
• Level 2: Classify \( 5x^3 - 4x + 1 \) as monomial, binomial, or trinomial.
• Level 3: Determine if \( 2x^{1/2} + 3 \) is a polynomial and justify.

Answer Key

• Level 1: Yes, it is a polynomial of degree 4.
• Level 2: Trinomial (three terms).
• Level 3: Not a polynomial because exponent is not a whole number.

Quick Reference

• Polynomial: sum of terms with variables raised to whole number exponents.
• Monomial: one term.
• Binomial: two terms.
• Trinomial: three terms.

Glossary

• Polynomial: Algebraic expression with non-negative integer exponents.
• Degree: Highest exponent of variable in polynomial.
• Monomial: Single term polynomial.
• Binomial: Polynomial with two terms.
• Trinomial: Polynomial with three terms.

Standard Form of Polynomials

The standard form of a polynomial arranges its terms in descending order of degree. For example, a polynomial \( p(x) \) of degree \( n \) is written as:

\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]

where \( a_n \neq 0 \).

Worked Example

Write the polynomial \( 3 + 4x^3 - 2x + x^2 \) in standard form.

Solution:

Arrange terms from highest to lowest degree:

\[ 4x^3 + x^2 - 2x + 3 \]

Practice Set

• Level 1: Write \( 5x + 2x^2 + 7 \) in standard form.
• Level 2: Arrange \( 6 - 3x^4 + x^3 \) in standard form.
• Level 3: Express \( x - 4 + 2x^5 - 3x^2 \) in standard form.

Answer Key

• Level 1: \( 2x^2 + 5x + 7 \)
• Level 2: \( -3x^4 + x^3 + 6 \)
• Level 3: \( 2x^5 - 3x^2 + x - 4 \)

Quick Reference

• Standard form: terms ordered by descending powers of \( x \).

Glossary

• Standard form: Polynomial with terms arranged from highest to lowest degree.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression.

For example, in \( p(x) = 4x^5 + 3x^3 - x + 7 \), the degree is 5.

Significance

• Determines the shape and behavior of the polynomial graph.
• Indicates the maximum number of zeros the polynomial can have.

Worked Example

Find the degree of \( 2x^4 - 5x^2 + 3x - 1 \).

Solution: The highest power of \( x \) is 4, so degree is 4.

Practice Set

• Level 1: Degree of \( 7x^3 + 2x + 1 \).
• Level 2: Degree of \( 5x^7 - 3x^5 + x^2 \).
• Level 3: Degree of \( 4x^0 + 6x^1 - 2x^3 + x^4 \).

Answer Key

• Level 1: 3
• Level 2: 7
• Level 3: 4

Quick Reference

• Degree = highest exponent of variable.

Glossary

• Degree: Highest power of variable in polynomial.

Types of Polynomials Based on Degree

Polynomials are classified by their degree as follows:

• Linear Polynomial: Degree 1, e.g., \( ax + b \).
• Quadratic Polynomial: Degree 2, e.g., \( ax^2 + bx + c \).
• Cubic Polynomial: Degree 3, e.g., \( ax^3 + bx^2 + cx + d \).

Worked Example

Classify \( 3x^3 + 2x^2 - x + 5 \).

Solution: Degree is 3, so it is a cubic polynomial.

Practice Set

• Level 1: Identify the type of \( 4x + 7 \).
• Level 2: Identify the type of \( 5x^2 - 3x + 1 \).
• Level 3: Identify the type of \( 2x^4 + x^3 - 5 \).

Answer Key

• Level 1: Linear polynomial.
• Level 2: Quadratic polynomial.
• Level 3: Polynomial of degree 4 (quartic).

Quick Reference

• Linear: degree 1
• Quadratic: degree 2
• Cubic: degree 3
• Quartic: degree 4

Glossary

• Linear Polynomial: Degree 1 polynomial.
• Quadratic Polynomial: Degree 2 polynomial.
• Cubic Polynomial: Degree 3 polynomial.

Polynomial Operations

Operations on polynomials include addition, subtraction, multiplication, and division.

Addition and Subtraction

Combine like terms (terms with the same variable and exponent).

Example: Add \( (3x^2 + 2x + 1) + (5x^2 - x + 4) \)

Step 1: Group like terms:

\( (3x^2 + 5x^2) + (2x - x) + (1 + 4) \)

Step 2: Add coefficients:

\( 8x^2 + x + 5 \)

Multiplication

Multiply each term of the first polynomial by each term of the second polynomial and combine like terms.

Example: Multiply \( (x + 2)(x + 3) \)

Step 1: Multiply terms:

\( x \times x = x^2 \)

\( x \times 3 = 3x \)

\( 2 \times x = 2x \)

\( 2 \times 3 = 6 \)

Step 2: Add like terms:

\( x^2 + 3x + 2x + 6 = x^2 + 5x + 6 \)

Division

Polynomial division is performed similar to long division of numbers.

Practice Set

• Level 1: Add \( (2x^2 + 3x) + (x^2 - x + 4) \).
• Level 2: Subtract \( (5x^3 + 2x) - (3x^3 + x) \).
• Level 3: Multiply \( (x + 4)(x^2 - x + 1) \).

Answer Key

• Level 1: \( 3x^2 + 2x + 4 \)
• Level 2: \( 2x^3 + x \)
• Level 3: \( x^3 - x^2 + x + 4x^2 - 4x + 4 = x^3 + 3x^2 - 3x + 4 \)

Quick Reference

• Add/subtract by combining like terms.
• Multiply each term by every term in the other polynomial.
• Divide using polynomial long division.

Glossary

• Like terms: Terms with the same variable and exponent.
• Polynomial multiplication: Distributive multiplication of terms.
• Polynomial division: Dividing polynomials similar to numbers.

Factorisation of Polynomials

Factorisation is expressing a polynomial as a product of its factors.

Common Methods

• Factor by grouping: Group terms and factor common factors.
• Use of identities: Apply formulas like \( a^2 - b^2 = (a - b)(a + b) \).
• Quadratic factorisation: For quadratic \( ax^2 + bx + c \), find factors of \( ac \) that sum to \( b \).

Worked Example

Factorise \( x^2 + 5x + 6 \).

Step 1: Find two numbers that multiply to 6 and add to 5: 2 and 3.

Step 2: Write as \( x^2 + 2x + 3x + 6 \).

Step 3: Group terms: \( (x^2 + 2x) + (3x + 6) \).

Step 4: Factor each group: \( x(x + 2) + 3(x + 2) \).

Step 5: Factor common binomial: \( (x + 2)(x + 3) \).

Practice Set

• Level 1: Factorise \( x^2 - 9 \).
• Level 2: Factorise \( x^2 + 7x + 12 \).
• Level 3: Factorise \( 2x^2 + 7x + 3 \).

Answer Key

• Level 1: \( (x - 3)(x + 3) \)
• Level 2: \( (x + 3)(x + 4) \)
• Level 3: \( (2x + 1)(x + 3) \)

Quick Reference

• Factor by grouping or use identities.
• For quadratics, find factors of \( ac \) summing to \( b \).

Glossary

• Factorisation: Expressing polynomial as product of factors.
• Factor theorem: If \( f(a) = 0 \), then \( x - a \) is a factor.

Factor Theorem

The Factor Theorem states that for a polynomial \( f(x) \), if \( f(a) = 0 \), then \( (x - a) \) is a factor of \( f(x) \).

Proof

By polynomial division, dividing \( f(x) \) by \( (x - a) \) gives:

\[ f(x) = (x - a)q(x) + r \]

where \( q(x) \) is the quotient and \( r \) is the remainder (a constant).

Substitute \( x = a \):

\[ f(a) = (a - a)q(a) + r = 0 + r = r \]

If \( f(a) = 0 \), then \( r = 0 \), so \( (x - a) \) divides \( f(x) \) exactly.

Worked Example

Check if \( (x - 2) \) is a factor of \( f(x) = x^3 - 4x^2 + 5x - 2 \).

Calculate \( f(2) \):

\( 2^3 - 4(2)^2 + 5(2) - 2 = 8 - 16 + 10 - 2 = 0 \)

Since \( f(2) = 0 \), \( (x - 2) \) is a factor.

Practice Set

• Level 1: Verify if \( (x - 1) \) is a factor of \( x^2 - 3x + 2 \).
• Level 2: Check if \( (x + 3) \) is a factor of \( x^3 + 2x^2 - 9x - 18 \).
• Level 3: Determine if \( (x - 4) \) is a factor of \( 2x^3 - 3x^2 - 8x + 12 \).

Answer Key

• Level 1: \( f(1) = 1 - 3 + 2 = 0 \), so factor.
• Level 2: \( f(-3) = -27 + 18 + 27 - 18 = 0 \), so factor.
• Level 3: \( f(4) = 128 - 48 - 32 + 12 = 60 \neq 0 \), not a factor.

Quick Reference

• If \( f(a) = 0 \), then \( (x - a) \) is a factor.

Glossary

• Factor Theorem: Connects zeros of polynomial to factors.

Zeros of Polynomials

A zero (root) of a polynomial \( p(x) \) is a value \( k \) such that \( p(k) = 0 \).

Geometrically, zeros correspond to x-intercepts of the graph of \( y = p(x) \).

Properties:

• A polynomial of degree \( n \) has at most \( n \) zeros.
• Zeros can be real or complex.

Worked Example

Find zeros of \( p(x) = x^2 - 5x + 6 \).

Factorise:

\( x^2 - 5x + 6 = (x - 2)(x - 3) \)

Zeros are \( x = 2 \) and \( x = 3 \).

Practice Set

• Level 1: Find zeros of \( x^2 - 4 \).
• Level 2: Find zeros of \( x^2 + 3x + 2 \).
• Level 3: Find zeros of \( 2x^3 - 3x^2 - 2x + 3 \).

Answer Key

• Level 1: \( x = 2, -2 \)
• Level 2: \( x = -1, -2 \)
• Level 3: \( x = 1, \frac{3}{2}, -1 \)

Quick Reference

• Zeros satisfy \( p(x) = 0 \).
• Number of zeros ≤ degree of polynomial.

Glossary

• Zero of polynomial: Value where polynomial equals zero.

Graphing Polynomials

The graph of a polynomial \( p(x) \) is a curve in the coordinate plane.

Key features:

• Degree determines the general shape.
• Leading coefficient affects end behavior.
• Zeros correspond to x-intercepts.
• For quadratic polynomials \( ax^2 + bx + c \), the graph is a parabola.

Quadratic Graph Details

Vertex: \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \)

Axis of symmetry: \( x = -\frac{b}{2a} \)

Direction: Opens upward if \( a > 0 \), downward if \( a < 0 \).

Discriminant \( D = b^2 - 4ac \)

• \( D > 0 \): Two distinct real roots, parabola crosses x-axis twice.
• \( D = 0 \): One real root, parabola touches x-axis.
• \( D < 0 \): No real roots, parabola does not intersect x-axis.

Worked Example

Graph \( y = x^2 - 4x + 3 \).

Calculate vertex:

\( x = -\frac{-4}{2 \times 1} = 2 \)

\( y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \)

Vertex at \( (2, -1) \), parabola opens upward.

Discriminant:

\( D = (-4)^2 - 4(1)(3) = 16 - 12 = 4 > 0 \)

Two distinct real roots.

Practice Set

• Level 1: Find vertex and axis of symmetry of \( y = x^2 + 6x + 8 \).
• Level 2: Determine number of roots of \( y = 2x^2 - 4x + 2 \).
• Level 3: Sketch graph of \( y = -x^2 + 4x - 3 \).

Answer Key

• Level 1: Vertex at \( (-3, -1) \), axis \( x = -3 \).
• Level 2: \( D = (-4)^2 - 4(2)(2) = 16 - 16 = 0 \), one real root.
• Level 3: Parabola opens downward, vertex at \( (2, 1) \).

Quick Reference

• Vertex formula: \( x = -\frac{b}{2a} \).
• Discriminant determines roots.
• Leading coefficient sign determines parabola direction.

Glossary

• Vertex: Highest or lowest point of parabola.
• Axis of symmetry: Vertical line through vertex.
• Discriminant: Determines nature of roots.

Applications of Polynomials

Polynomials model various real-world phenomena in physics, engineering, economics, and other fields.

• Physics: Motion equations, projectile paths.
• Engineering: Signal processing, control systems.
• Economics: Cost and revenue functions.
• Traffic flow: Modeling and optimization.

Worked Example

In economics, profit \( P(x) \) can be modeled as a polynomial \( P(x) = -2x^2 + 40x - 100 \), where \( x \) is the number of units sold.

Find the number of units \( x \) that maximizes profit.

Step 1: Find vertex \( x = -\frac{b}{2a} = -\frac{40}{2 \times (-2)} = 10 \).

Step 2: Maximum profit at \( x = 10 \) units.

Practice Set

• Level 1: Identify degree of polynomial modeling cost \( C(x) = 5x + 100 \).
• Level 2: Find zeros of \( R(x) = x^2 - 9 \) representing revenue.
• Level 3: For \( P(x) = -3x^2 + 12x - 9 \), find maximum profit and units sold.

Answer Key

• Level 1: Degree 1 (linear).
• Level 2: Zeros at \( x = 3, -3 \).
• Level 3: Vertex at \( x = 2 \), maximum profit \( P(2) = -3(4) + 24 - 9 = 3 \).

Quick Reference

• Polynomials model real-world relationships.
• Vertex gives maxima or minima in quadratic models.

Glossary

• Application: Use of polynomials to model real situations.
• Maximum/Minimum: Highest or lowest value of polynomial function.