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Updated 20 Feb 2026

•

11 pages

Comprehensive Algebra 2 Study Guide

Hannah Watters

@hannahwatters

Ever struggled with algebra and wondered when you'll actually use... Show more

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ALGEBRA 2
The Factor Theorem

* If ox=a is a root then ox-a is a foctor...

a
Eg: If ox-7 is a factor of ox³-3x²-25x-21 find the roots of th

The Factor Theorem - Finding Roots

The Factor Theorem is your shortcut to solving cubic equations without guessing. If x = a is a root, then $x - a$ is a factor - it's that simple!

When you know one factor, use long division to break down the cubic into a quadratic you can easily solve. For example, if $x - 7$ is a factor of x³ - 3x² - 25x - 21, divide it out to get x² + 4x + 3, then factorise normally.

Here's a handy trick: all roots will divide evenly into the constant term (the number without x). So if your roots are 7, -1, and -3, they'll all divide into 21 with no remainder. This is brilliant for checking your answers!

Quick Check: Always verify your roots divide into the constant term - it's a foolproof way to spot mistakes.

Finding Missing Coefficients

Sometimes you'll know the factors but need to find the missing coefficients - don't panic, it's actually quite straightforward! Start by substituting the known roots into the equation to create simultaneous equations.

If $x + 2$ and $x - 3$ are factors of 3x³ + ax² + bx - 24, substitute x = -2 and x = 3 into the equation. This gives you two equations with a and b that you can solve simultaneously.

Once you've found a and b, use long division with one of the known factors to get a quadratic. Then you can find the final root by factoring or using the quadratic formula.

Pro Tip: Always double-check by substituting your values back into the original equation - if it equals zero, you're spot on!

Solving the Quadratic

After using long division, you'll have a quadratic equation that's much easier to handle. Factor it using your usual methods - look for two numbers that multiply to give ac and add to give b.

In our example, 3x² - 5x - 12 = 0 factors to $3x + 4$ $x - 3$ = 0. This gives us x = -4/3 and x = 3. Combined with our known root x = -2, we have all three roots: 3, -2, and -4/3.

The beauty of this method is that it breaks down intimidating cubic equations into manageable chunks. You're essentially doing what you already know - just with an extra step at the beginning.

Remember: Every cubic equation has exactly three roots (counting repeated roots), so you know when you're finished!

Completing the Square - The Basics

Completing the square transforms any quadratic into the form $x + p$ ² + q, which instantly reveals the turning point of the graph. This technique is essential for understanding parabolas and solving optimization problems.

For x² + 4x - 5, take half the coefficient of x (which is 4), square it to get 4, then adjust: $x + 2$ ² - 9. The turning point is at (-2, -9) - just change the sign of p and read off q.

The process is systematic: half the x coefficient, square it, add and subtract this value, then rearrange. With x² + 7x + 9, you get $x + 7/2$ ² - 13/4, giving a turning point at (-7/2, -13/4).

Key Insight: The turning point coordinates come directly from the completed square form - no extra calculations needed!

Completing the Square with Coefficients

When the coefficient of x² isn't 1, don't worry - just factor it out first, complete the square inside the brackets, then multiply back through.

For 2x² + 8x - 9, factor out the 2 to get 2 $x² + 4x - 9/2$ . Complete the square inside: 2 $(x + 2)² - 17/2$ . Finally, multiply the 2 back through: 2 $x + 2$ ² - 17.

The same method works for any coefficient. With 3x² - x + 6, you get 3 $x - 1/6$ ² + 71/12. The key is staying organized and working step by step.

Stay Organized: Write each step clearly - it's easy to make sign errors when working with fractions!

The Discriminant and Types of Roots

The discriminant $b² - 4ac$ tells you everything about the roots before you even solve the equation. It's like having x-ray vision for quadratics!

When b² - 4ac > 0, you get two distinct real roots - the parabola crosses the x-axis twice. When it equals 0, you have two equal real roots $the parabola just touches the x-axis$ . When it's negative, there are no real roots - the parabola doesn't touch the x-axis at all.

For example, 3x² - 6x - 1 has discriminant 48 > 0, so two different real roots. Meanwhile, x² - 2x + 10 has discriminant -36 < 0, so no real solutions.

Visual Tip: Think of the discriminant as predicting how many times your parabola will cross the x-axis!

Using the Discriminant to Find Values

You can work backwards with the discriminant to find unknown coefficients. If you're told an equation has specific types of roots, set up the discriminant condition and solve.

For non-real roots, you need b² - 4ac < 0. With x² - $k-4$ x + 1 = 0, this gives $k-4$ ² - 4 < 0, which simplifies to k² - 8k + 12 < 0. Factoring gives 2 < k < 6.

This technique is particularly useful in exam questions where you need to find ranges of values. The discriminant gives you the inequality, then you solve it like any other quadratic inequality.

Exam Strategy: These "find the range" questions often appear on papers - master the discriminant method and they become straightforward!

Graphing Cubic Polynomials

Cubic polynomial graphs have characteristic shapes that depend on the sign of the leading coefficient. Positive x³ terms create graphs that go from bottom-left to top-right, while negative x³ terms go from top-left to bottom-right.

The roots are where the graph crosses the x-axis. A single root means the graph crosses once, while a double root means it touches and bounces off. A triple root means the graph flattens out at that point.

For f(x) = x³ + 3x² - x - 3, the graph intersects at x = -3, -1, 1. For g(x) with factors $x-2$ $x-4$ ², it crosses at x = 2 and touches at x = 4 (double root).

Visual Learning: Sketch these graphs - seeing the relationship between factors and crossings makes everything click!

Even Degree Polynomials

Even degree polynomials (degree 2, 4, 6, etc.) have symmetrical behavior - both ends of the graph point in the same direction. The leading coefficient determines whether both arms point up (positive) or down (negative).

With positive leading coefficients like 2x⁴ + x - 6, both ends of the graph go upward to infinity. With negative leading coefficients like -2x² + x + 6, both ends go downward.

This is completely different from odd degree polynomials, where the ends go in opposite directions. Understanding this helps you sketch graphs quickly and predict behavior.

Memory Aid: Even degree = even behavior (both arms the same), odd degree = opposite arms!

Constructing Polynomial Expressions

When you're given information about roots and points on a graph, you can construct the polynomial expression by working backwards from the factors.

If there's a triple root at x = -1 and a single root at x = 1, start with $x + 1$ ³ $x - 1$ . For four identical roots at x = 3, you'd have $x - 3$ ⁴. Combine these based on what you're told.

Use the given point to find any missing coefficient. If (0, -81) lies on the graph and your expression is p $x + 1$ ³ $x - 1$ $x - 3$ ⁴, substitute to get -81 = p(-81), so p = 1.

Strategy: Build the expression step by step from the roots, then use the given point to nail down any unknowns!

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Mathematics

•

Updated 20 Feb 2026

•

11 pages

Comprehensive Algebra 2 Study Guide

Hannah Watters

@hannahwatters

Ever struggled with algebra and wondered when you'll actually use it? These concepts are essential for your Leaving Cert and will help you solve complex problems with confidence. From finding roots using the Factor Theorem to graphing polynomials, these skills... Show more

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The Factor Theorem - Finding Roots

The Factor Theorem is your shortcut to solving cubic equations without guessing. If x = a is a root, then $x - a$ is a factor - it's that simple!

Quick Check: Always verify your roots divide into the constant term - it's a foolproof way to spot mistakes.

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Finding Missing Coefficients

If $x + 2$ and $x - 3$ are factors of 3x³ + ax² + bx - 24, substitute x = -2 and x = 3 into the equation. This gives you two equations with a and b that you can solve simultaneously.

Once you've found a and b, use long division with one of the known factors to get a quadratic. Then you can find the final root by factoring or using the quadratic formula.

Pro Tip: Always double-check by substituting your values back into the original equation - if it equals zero, you're spot on!

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Solving the Quadratic

After using long division, you'll have a quadratic equation that's much easier to handle. Factor it using your usual methods - look for two numbers that multiply to give ac and add to give b.

In our example, 3x² - 5x - 12 = 0 factors to $3x + 4$ $x - 3$ = 0. This gives us x = -4/3 and x = 3. Combined with our known root x = -2, we have all three roots: 3, -2, and -4/3.

The beauty of this method is that it breaks down intimidating cubic equations into manageable chunks. You're essentially doing what you already know - just with an extra step at the beginning.

Remember: Every cubic equation has exactly three roots (counting repeated roots), so you know when you're finished!

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Completing the Square - The Basics

For x² + 4x - 5, take half the coefficient of x (which is 4), square it to get 4, then adjust: $x + 2$ ² - 9. The turning point is at (-2, -9) - just change the sign of p and read off q.

The process is systematic: half the x coefficient, square it, add and subtract this value, then rearrange. With x² + 7x + 9, you get $x + 7/2$ ² - 13/4, giving a turning point at (-7/2, -13/4).

Key Insight: The turning point coordinates come directly from the completed square form - no extra calculations needed!

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Completing the Square with Coefficients

When the coefficient of x² isn't 1, don't worry - just factor it out first, complete the square inside the brackets, then multiply back through.

For 2x² + 8x - 9, factor out the 2 to get 2 $x² + 4x - 9/2$ . Complete the square inside: 2 $(x + 2)² - 17/2$ . Finally, multiply the 2 back through: 2 $x + 2$ ² - 17.

The same method works for any coefficient. With 3x² - x + 6, you get 3 $x - 1/6$ ² + 71/12. The key is staying organized and working step by step.

Stay Organized: Write each step clearly - it's easy to make sign errors when working with fractions!

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The Discriminant and Types of Roots

The discriminant $b² - 4ac$ tells you everything about the roots before you even solve the equation. It's like having x-ray vision for quadratics!

For example, 3x² - 6x - 1 has discriminant 48 > 0, so two different real roots. Meanwhile, x² - 2x + 10 has discriminant -36 < 0, so no real solutions.

Visual Tip: Think of the discriminant as predicting how many times your parabola will cross the x-axis!

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Using the Discriminant to Find Values

You can work backwards with the discriminant to find unknown coefficients. If you're told an equation has specific types of roots, set up the discriminant condition and solve.

For non-real roots, you need b² - 4ac < 0. With x² - $k-4$ x + 1 = 0, this gives $k-4$ ² - 4 < 0, which simplifies to k² - 8k + 12 < 0. Factoring gives 2 < k < 6.

This technique is particularly useful in exam questions where you need to find ranges of values. The discriminant gives you the inequality, then you solve it like any other quadratic inequality.

Exam Strategy: These "find the range" questions often appear on papers - master the discriminant method and they become straightforward!

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Graphing Cubic Polynomials

For f(x) = x³ + 3x² - x - 3, the graph intersects at x = -3, -1, 1. For g(x) with factors $x-2$ $x-4$ ², it crosses at x = 2 and touches at x = 4 (double root).

Visual Learning: Sketch these graphs - seeing the relationship between factors and crossings makes everything click!

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Even Degree Polynomials

With positive leading coefficients like 2x⁴ + x - 6, both ends of the graph go upward to infinity. With negative leading coefficients like -2x² + x + 6, both ends go downward.

This is completely different from odd degree polynomials, where the ends go in opposite directions. Understanding this helps you sketch graphs quickly and predict behavior.

Memory Aid: Even degree = even behavior (both arms the same), odd degree = opposite arms!

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Constructing Polynomial Expressions

When you're given information about roots and points on a graph, you can construct the polynomial expression by working backwards from the factors.

If there's a triple root at x = -1 and a single root at x = 1, start with $x + 1$ ³ $x - 1$ . For four identical roots at x = 3, you'd have $x - 3$ ⁴. Combine these based on what you're told.

Use the given point to find any missing coefficient. If (0, -81) lies on the graph and your expression is p $x + 1$ ³ $x - 1$ $x - 3$ ⁴, substitute to get -81 = p(-81), so p = 1.

Strategy: Build the expression step by step from the roots, then use the given point to nail down any unknowns!