IB Mathematics AA – Topic 2: Functions

Three Related Concepts:

• Zero: A value \(a\) where \(P(a) = 0\)

  The y-coordinate becomes zero

• Root: A solution to the equation \(P(x) = 0\)

  Same as a zero—the terms are interchangeable

• Factor: If \(a\) is a root, then \((x - a)\) is a factor

  The polynomial is divisible by \((x - a)\) with no remainder

Fundamental Relationship:

Understanding Repeated Roots:

• Multiplicity 1 (simple root): Graph crosses x-axis at this point
• Multiplicity 2 (double root): Graph touches x-axis but doesn't cross (turning point)
• Multiplicity 3 (triple root): Graph crosses with flattened appearance
• General rule: Even multiplicity → touches axis; Odd multiplicity → crosses axis

Example: \(P(x) = (x-2)^2(x+1)\) has a double root at \(x=2\) and simple root at \(x=-1\)

Remainder Theorem

This allows us to find remainders without performing long division!

Factor Theorem

Common Uses:

• Finding remainders: Evaluate \(P(a)\) instead of dividing
• Testing factors: Check if \(P(a) = 0\) to verify \((x-a)\) is a factor
• Finding unknown coefficients: If you know a factor, use \(P(a) = 0\) to solve for unknowns
• Factoring completely: Find one factor, divide, then factor the quotient

⚠ Common Pitfalls:

• Sign errors: For \((x+3)\), test \(x = -3\), not \(x = 3\)!
• Forgetting division: After finding one factor, must divide to get remaining factors
• Calculation mistakes: Carefully substitute, especially with negative values and powers
• Not checking all factors: Test systematically—factors of constant term over factors of leading coefficient

Example 1: Using Factor and Remainder Theorems

Problem: Consider \(P(x) = 2x^3 - 5x^2 - 4x + 3\)

(a) Find the remainder when \(P(x)\) is divided by \((x-2)\)

(b) Show that \((x-3)\) is a factor of \(P(x)\)

(c) Hence, factorize \(P(x)\) completely

Solution:

(a) Remainder when divided by \((x-2)\):

By Remainder Theorem, remainder = \(P(2)\)

\(P(2) = 2(2)^3 - 5(2)^2 - 4(2) + 3\)

\(= 2(8) - 5(4) - 8 + 3\)

\(= 16 - 20 - 8 + 3\)

\(= -9\)

Remainder: -9

(b) Show \((x-3)\) is a factor:

By Factor Theorem, \((x-3)\) is a factor if \(P(3) = 0\)

\(P(3) = 2(3)^3 - 5(3)^2 - 4(3) + 3\)

\(= 2(27) - 5(9) - 12 + 3\)

\(= 54 - 45 - 12 + 3\)

\(= 0\) ✓

Since \(P(3) = 0\), \((x-3)\) is a factor

(c) Complete factorization:

Since \((x-3)\) is a factor, divide \(P(x)\) by \((x-3)\):

Using polynomial division or synthetic division:

\(P(x) = (x-3)(2x^2 + x - 1)\)

Factor the quadratic: \(2x^2 + x - 1 = (2x-1)(x+1)\)

\(P(x) = (x-3)(2x-1)(x+1)\)

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

For \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\):

Example: For \(3x^2 - 6x + 2 = 0\): sum = \(\frac{6}{3} = 2\), product = \(\frac{2}{3}\)

For \(ax^3 + bx^2 + cx + d = 0\) with roots \(\alpha, \beta, \gamma\):

Common Uses:

• Finding unknowns: If you know the roots, find coefficients
• Checking work: Verify your roots satisfy the sum and product
• Forming equations: Given roots, construct the polynomial
• Finding related expressions: Calculate \(\alpha^2 + \beta^2\), \(\frac{1}{\alpha} + \frac{1}{\beta}\), etc.

💡 Useful Relationships:

• \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
• \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}\)
• \((\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\)
• \(\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)\)

Example 2: Sum and Product of Roots (IB-Style)

Problem: The equation \(x^2 - 3x + k = 0\) has roots \(\alpha\) and \(\beta\)

(a) Express \(\alpha + \beta\) and \(\alpha\beta\) in terms of \(k\)

(b) Given that \(\alpha^2 + \beta^2 = 13\), find the value of \(k\)

(c) Hence find the values of \(\alpha\) and \(\beta\)

Solution:

(a) Sum and product:

For \(x^2 - 3x + k = 0\): \(a = 1, b = -3, c = k\)

\(\alpha + \beta = -\frac{b}{a} = -\frac{-3}{1} = 3\)

\(\alpha\beta = \frac{c}{a} = \frac{k}{1} = k\)

(b) Finding \(k\):

Use the identity: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)

Substitute known values:

\(13 = (3)^2 - 2k\)

\(13 = 9 - 2k\)

\(2k = 9 - 13 = -4\)

\(k = -2\)

(c) Finding \(\alpha\) and \(\beta\):

Equation becomes: \(x^2 - 3x - 2 = 0\)

Using quadratic formula:

\(x = \frac{3 \pm \sqrt{9 + 8}}{2} = \frac{3 \pm \sqrt{17}}{2}\)

\(\alpha = \frac{3 + \sqrt{17}}{2}\), \(\beta = \frac{3 - \sqrt{17}}{2}\)

Check: \(\alpha + \beta = 3\) ✓ and \(\alpha\beta = \frac{9-17}{4} = -2\) ✓

Step-by-Step Factoring:

1. Check for common factors: Factor out greatest common factor
2. Use Rational Root Theorem: Test factors of constant term ÷ factors of leading coefficient
3. Apply Factor Theorem: Find one root by testing, then you have a factor
4. Polynomial division: Divide by known factor to reduce degree
5. Repeat or use formula: Factor remaining polynomial (quadratic formula if degree 2)
6. Write in factored form: Express as product of all factors

Finding Possible Rational Roots:

Example: For \(2x^3 - 3x^2 - 5x + 6\), possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}\)

Example 3: Complete Polynomial Factorization

Problem: Given \(P(x) = x^3 - 6x^2 + 11x - 6\)

(a) Find all zeros

(b) Express in fully factored form

(c) Verify using sum and product of roots

Solution:

(a) Finding zeros:

Possible rational roots (by Rational Root Theorem): \(\pm 1, \pm 2, \pm 3, \pm 6\)

Test \(x = 1\): \(P(1) = 1 - 6 + 11 - 6 = 0\) ✓

So \((x-1)\) is a factor

Divide \(P(x)\) by \((x-1)\):

\(P(x) = (x-1)(x^2 - 5x + 6)\)

Factor the quadratic: \(x^2 - 5x + 6 = (x-2)(x-3)\)

Zeros: \(x = 1, 2, 3\)

(b) Factored form:

\(P(x) = (x-1)(x-2)(x-3)\)

(c) Verification using Vieta's formulas:

For \(x^3 - 6x^2 + 11x - 6\) with roots \(\alpha = 1, \beta = 2, \gamma = 3\):

Sum of roots:

\(\alpha + \beta + \gamma = 1 + 2 + 3 = 6 = -\frac{-6}{1}\) ✓

Sum of products of pairs:

\(\alpha\beta + \alpha\gamma + \beta\gamma = 2 + 3 + 6 = 11 = \frac{11}{1}\) ✓

Product of roots:

\(\alpha\beta\gamma = 1 \times 2 \times 3 = 6 = -\frac{-6}{1}\) ✓

All relationships verified ✓

Common Question Types:

• "Find the remainder": Use Remainder Theorem—evaluate \(P(a)\)
• "Show that (x-k) is a factor": Use Factor Theorem—prove \(P(k) = 0\)
• "Hence factorize completely": Divide by known factor, then factor quotient
• "Given sum/product of roots": Use Vieta's formulas to find unknowns
• "Find all zeros": Use Rational Root Theorem, test systematically

Key Reminders:

• Always check your factorization by expanding
• For \((x+a)\), test \(x = -a\) (negative!)
• Use GDC to verify roots and graph polynomial
• Show all algebraic steps—don't skip work