Polynomials

Overview

Polynomials form a foundational topic in upper-primary mathematics, bridging arithmetic and advanced algebra. For KAR TET Paper II, you must understand what polynomials are, how to find their zeros, and how to apply the division algorithm—skills essential for teaching Classes 6–8 students. This topic directly connects to factorisation, quadratic equations, and algebraic identities in the syllabus.

Expect questions testing your ability to identify polynomial types, find zeros graphically and algebraically, verify relationships between zeros and coefficients, and perform polynomial division. Mastery here also supports pedagogy questions on teaching algebraic thinking to young learners.

Key Concepts

• **Polynomial definition**: An algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where coefficients are real numbers and exponents are whole numbers (non-negative integers).

• **Degree of a polynomial**: The highest power of the variable with a non-zero coefficient. Example: 3x⁴ + 2x – 7 has degree 4.

• **Types by degree**: Constant (degree 0), Linear (degree 1), Quadratic (degree 2), Cubic (degree 3), Biquadratic/Quartic (degree 4).

• **Zero/Root of a polynomial**: A value 'a' such that p(a) = 0. Geometrically, zeros are the x-coordinates where the graph crosses or touches the x-axis.

• **Number of zeros**: A polynomial of degree n has at most n real zeros. A linear polynomial has exactly 1 zero; a quadratic has at most 2; a cubic has at most 3.

• **Relationship between zeros and coefficients**: For quadratic ax² + bx + c with zeros α and β: Sum (α + β) = –b/a; Product (αβ) = c/a.

• **Division Algorithm**: For polynomials p(x) and g(x) where g(x) ≠ 0, there exist unique q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where degree of r(x) < degree of g(x) or r(x) = 0.

• **Factor Theorem**: (x – a) is a factor of p(x) if and only if p(a) = 0.

Formulas / Key Facts

| Formula/Fact | Context | |--------------|---------| | p(a) = 0 implies 'a' is a zero | Definition of zero/root | | Linear p(x) = ax + b has zero at x = –b/a | Finding zero of linear polynomial | | Quadratic: α + β = –b/a | Sum of zeros for ax² + bx + c | | Quadratic: αβ = c/a | Product of zeros for ax² + bx + c | | Cubic: α + β + γ = –b/a | Sum of zeros for ax³ + bx² + cx + d | | Cubic: αβ + βγ + γα = c/a | Sum of products taken two at a time | | Cubic: αβγ = –d/a | Product of all three zeros | | Division Algorithm: Dividend = Divisor × Quotient + Remainder | Always applies; remainder degree < divisor degree | | Maximum real zeros = Degree of polynomial | Upper bound on number of zeros |

Worked Examples

**Example 1: Finding zeros of a quadratic polynomial**

Find the zeros of p(x) = x² – 5x + 6.

*Solution:* Step 1: Set p(x) = 0 → x² – 5x + 6 = 0 Step 2: Factorise → (x – 2)(x – 3) = 0 Step 3: Solve → x = 2 or x = 3

Zeros are 2 and 3.

*Verification:* Sum = 2 + 3 = 5 = –(–5)/1 = –b/a ✓ Product = 2 × 3 = 6 = 6/1 = c/a ✓

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**Example 2: Forming a quadratic polynomial from given zeros**

Form a quadratic polynomial with zeros 4 and –3.

*Solution:* Sum of zeros (α + β) = 4 + (–3) = 1 Product of zeros (αβ) = 4 × (–3) = –12

Polynomial = x² – (sum)x + (product) = x² – 1x + (–12) = **x² – x – 12**

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**Example 3: Division Algorithm**

Divide p(x) = 3x³ + x² + 2x + 5 by g(x) = x + 1.

*Solution:* Using long division:

``` 3x² – 2x + 4 ________________ x + 1 | 3x³ + x² + 2x + 5 3x³ + 3x² ___________ –2x² + 2x –2x² – 2x __________ 4x + 5 4x + 4 ______ 1 ```

Quotient q(x) = 3x² – 2x + 4 Remainder r(x) = 1

*Verification:* g(x) × q(x) + r(x) = (x + 1)(3x² – 2x + 4) + 1 = 3x³ – 2x² + 4x + 3x² – 2x + 4 + 1 = 3x³ + x² + 2x + 5 = p(x) ✓

Common Mistakes

• **Confusing zeros with coefficients**: Students think the constant term is a zero. *Correct thinking*: A zero is a value that makes the polynomial equal to zero, not simply a term in the expression.

• **Sign errors in sum of zeros**: Forgetting the negative sign in α + β = –b/a. *Fix*: Always write the formula with the negative sign first, then substitute.

• **Incomplete factorisation**: Stopping at partial factors like x(x – 3) = 0 and missing x = 0 as a zero. *Fix*: Set each factor equal to zero separately.

• **Ignoring remainder in division**: Assuming division is complete when remainder isn't zero or has degree greater than or equal to divisor. *Fix*: Continue until remainder's degree is strictly less than divisor's degree.

• **Treating expressions with negative exponents as polynomials**: Writing x⁻² + 3x as a polynomial. *Fix*: Polynomials require non-negative integer exponents only.

• **Miscounting degree**: Taking the first term's exponent without checking if coefficient is non-zero. *Fix*: Ensure the leading coefficient is not zero before stating the degree.

Quick Reference

• Polynomial = Expression with non-negative integer powers and real coefficients
• Degree = Highest power with non-zero coefficient
• Zero of p(x) → value where p(x) = 0
• Quadratic zeros: Sum = –b/a, Product = c/a
• Division Algorithm: p(x) = g(x) × q(x) + r(x), deg(r) < deg(g)
• Factor Theorem: (x – a) is factor ⟺ p(a) = 0