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Jan 28, 2026

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7 pages

Understanding the Rational Root Theorem

Shaivi Ramisetti

@shaivi711

Polynomial theorems like the Remainder, Factor, and Rational Root theorems... Show more

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Remainder Theorem

Ever wondered how to find the remainder when dividing a polynomial by a binomial? The Remainder Theorem makes this super easy! It states that when a polynomial function f(x) is divided by $x-c$ , the remainder equals f(c).

Let's see this in action. To find the remainder when dividing x⁴ + 8x³ + 12x² - 7x - 14 by x + 3, we calculate f(-3): f(-3) = (-3)⁴ + 8(-3)³ + 12(-3)² - 7(-3) - 14 f(-3) = 81 - 216 + 108 + 21 - 14 = -20

So the remainder is -20. No long division needed!

💡 Quick Tip: When the divisor is $x-c$ , just substitute c into the original polynomial to find the remainder!

Factor Theorem

The Factor Theorem connects zeros and factors of polynomials. It states that $x-c$ is a factor of polynomial f(x) if and only if f(c) = 0.

This means we can test whether a binomial is a factor by evaluating the polynomial at the potential zero. For example, with f(x) = x³ - 31x + 30, we can check if $x+1$ , $x-5$ , or $x+6$ are factors:

For $x+1$ : Calculate f(-1). If f(-1) = 0, then $x+1$ is a factor. For $x-5$ : Calculate f(5). Since f(5) = 0, $x-5$ is a factor! For $x+6$ : Calculate f(-6). Since f(-6) = 0, $x+6$ is also a factor!

🔍 Remember: When f(c) = 0, it means that c is a zero (or root) of the function and $x-c$ is a factor of the polynomial.

Rational Root Theorem

The Rational Root Theorem helps us find all possible rational roots of a polynomial with integer coefficients. This narrows down our search significantly!

For a polynomial f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ with integer coefficients, any rational root p/q (in lowest terms) must follow these rules:

p must be a factor of a₀ (the constant term)
q must be a factor of aₙ (the leading coefficient)

This means that instead of testing infinite values, we only need to check a finite list of possible rational roots.

⭐ Key Insight: This theorem doesn't tell you which values are actually roots - it just gives you a list of candidates to check using the Factor Theorem!

Applying the Rational Root Theorem

The Rational Root Theorem gives us a systematic way to find all possible rational roots of a polynomial. Here's how to use it:

First, identify the values for p and q:

p includes all positive and negative factors of the constant term
q includes all positive and negative factors of the leading coefficient

Then, create all possible fractions p/q in lowest form. These are your potential rational roots.

Remember that not all these candidates will be actual roots! You'll need to test each one using the Remainder Theorem or Factor Theorem to confirm.

🧩 Strategy Tip: Start by testing the simplest values (like ±1) first, since they're usually easier to calculate!

Finding Possible Rational Roots

Let's find all possible rational roots for two example polynomials.

For 4x³ - 4x² - x + 1:

p (factors of constant term 1): ±1
q (factors of leading coefficient 4): ±1, ±2, ±4
Possible rational roots: ±1, ±1/2, ±1/4

For 3x³ - x² - 18x + 16:

p (factors of constant term 16): ±1, ±2, ±4, ±8, ±16
q (factors of leading coefficient 3): ±1, ±3
Possible rational roots: ±1, ±1/3, ±2, ±2/3, ±4, ±4/3, ±8, ±8/3, ±16, ±16/3

🔢 Organization Tip: Make a systematic list of all your possible rational roots before you start testing them - this helps avoid missing any candidates!

Synthetic Division and Finding All Roots

Once we have our list of possible rational roots, we can use synthetic division to test them efficiently. When we find a root, we can divide the polynomial by $x-root$ to get a lower-degree polynomial.

For the polynomial f(x) = x³+3x²-6x-8, we can test roots like 1 and 2:

Testing x=2 with synthetic division gives a remainder of 0, so 2 is a root!
Dividing by $x-2$ gives us x²+5x+4, which factors as $x+4$ $x+1$

So the roots of the original polynomial are 2, -4, and -1.

Similarly, for f(x) = x³-21x+20, we find that 1 is a root, and the resulting quadratic x²+x-20 factors as $x+5$ $x-4$ .

✅ Check Your Work: After finding all roots, multiply the factors together to verify you get the original polynomial.

Solving Complex Polynomials

For higher-degree polynomials like f(x) = x⁴ + 3x³ - 7x² - 27x - 18, we can use the same process. First, list possible rational roots (±1, ±2, ±3, ±6, ±9, ±18), then test them systematically.

When we find that x=3 is a root, we use synthetic division to get a cubic polynomial: x³ + 6x² + 11x + 6. We continue the process:

Test x=-1: It's a root!
Dividing gives us x² + 5x + 6, which factors as $x+3$ $x+2$

So the roots of the original quartic polynomial are 3, -1, -3, and -2.

For another example, f(x) = 2x³-5x²-x+6, we find roots 3/2, -1, and 2 by applying the same methodical approach.

🏆 Master Strategy: Work systematically from one root to the next, reducing the polynomial's degree each time until you reach a quadratic that you can solve by factoring or the quadratic formula.

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Jan 28, 2026

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7 pages

Understanding the Rational Root Theorem

Shaivi Ramisetti

@shaivi711

Polynomial theorems like the Remainder, Factor, and Rational Root theorems give us powerful tools for analyzing polynomial functions. These theorems help us find factors, zeros, and remainders without using lengthy division or factoring methods.

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Remainder Theorem

So the remainder is -20. No long division needed!

💡 Quick Tip: When the divisor is $x-c$ , just substitute c into the original polynomial to find the remainder!

Sign up to see the contentIt's free!

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Improve your grades

Join milions of students

Factor Theorem

The Factor Theorem connects zeros and factors of polynomials. It states that $x-c$ is a factor of polynomial f(x) if and only if f(c) = 0.

For $x+1$ : Calculate f(-1). If f(-1) = 0, then $x+1$ is a factor. For $x-5$ : Calculate f(5). Since f(5) = 0, $x-5$ is a factor! For $x+6$ : Calculate f(-6). Since f(-6) = 0, $x+6$ is also a factor!

🔍 Remember: When f(c) = 0, it means that c is a zero (or root) of the function and $x-c$ is a factor of the polynomial.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Rational Root Theorem

The Rational Root Theorem helps us find all possible rational roots of a polynomial with integer coefficients. This narrows down our search significantly!

For a polynomial f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ with integer coefficients, any rational root p/q (in lowest terms) must follow these rules:

p must be a factor of a₀ (the constant term)
q must be a factor of aₙ (the leading coefficient)

This means that instead of testing infinite values, we only need to check a finite list of possible rational roots.

⭐ Key Insight: This theorem doesn't tell you which values are actually roots - it just gives you a list of candidates to check using the Factor Theorem!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Applying the Rational Root Theorem

The Rational Root Theorem gives us a systematic way to find all possible rational roots of a polynomial. Here's how to use it:

First, identify the values for p and q:

p includes all positive and negative factors of the constant term
q includes all positive and negative factors of the leading coefficient

Then, create all possible fractions p/q in lowest form. These are your potential rational roots.

Remember that not all these candidates will be actual roots! You'll need to test each one using the Remainder Theorem or Factor Theorem to confirm.

🧩 Strategy Tip: Start by testing the simplest values (like ±1) first, since they're usually easier to calculate!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Finding Possible Rational Roots

Let's find all possible rational roots for two example polynomials.

For 4x³ - 4x² - x + 1:

p (factors of constant term 1): ±1
q (factors of leading coefficient 4): ±1, ±2, ±4
Possible rational roots: ±1, ±1/2, ±1/4

For 3x³ - x² - 18x + 16:

p (factors of constant term 16): ±1, ±2, ±4, ±8, ±16
q (factors of leading coefficient 3): ±1, ±3
Possible rational roots: ±1, ±1/3, ±2, ±2/3, ±4, ±4/3, ±8, ±8/3, ±16, ±16/3

🔢 Organization Tip: Make a systematic list of all your possible rational roots before you start testing them - this helps avoid missing any candidates!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Synthetic Division and Finding All Roots

For the polynomial f(x) = x³+3x²-6x-8, we can test roots like 1 and 2:

Testing x=2 with synthetic division gives a remainder of 0, so 2 is a root!
Dividing by $x-2$ gives us x²+5x+4, which factors as $x+4$ $x+1$

So the roots of the original polynomial are 2, -4, and -1.

Similarly, for f(x) = x³-21x+20, we find that 1 is a root, and the resulting quadratic x²+x-20 factors as $x+5$ $x-4$ .

✅ Check Your Work: After finding all roots, multiply the factors together to verify you get the original polynomial.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Solving Complex Polynomials

When we find that x=3 is a root, we use synthetic division to get a cubic polynomial: x³ + 6x² + 11x + 6. We continue the process:

Test x=-1: It's a root!
Dividing gives us x² + 5x + 6, which factors as $x+3$ $x+2$

So the roots of the original quartic polynomial are 3, -1, -3, and -2.

For another example, f(x) = 2x³-5x²-x+6, we find roots 3/2, -1, and 2 by applying the same methodical approach.

🏆 Master Strategy: Work systematically from one root to the next, reducing the polynomial's degree each time until you reach a quadratic that you can solve by factoring or the quadratic formula.