Identifying Zeros of Polynomials

Understanding the zeros of a polynomial function is like finding the special points where the graph of the function crosses or touches the horizontal axis (the x-axis) 🎯. These points are also known as x-intercepts. For a polynomial function, f(x), a value of x is a zero if f(x) = 0.

Let’s break down how to identify and characterise these zeros from a graph or an equation.

Identifying Zeros of Polynomials 🕵️‍♀️

When you’re trying to find the zeros of a polynomial, you’re essentially looking for the x-values that make the function’s output zero. Here’s how you can find them:

• Factoring Technique ✂️: This is a crucial technique for finding zeros. If you can factor the polynomial function’s equation into a product of simpler expressions, you can then set each factor equal to zero and solve for x.
  • Greatest Common Factor (GCF): Sometimes, all terms in a polynomial share a common monomial factor, which can be factored out first.
  • Factoring by Grouping: For polynomials with four or more terms, you might group terms to find common factors.
  • Trinomial Factoring: For expressions resembling quadratic equations (e.g., ax² + bx + c or forms like t² - 8t + 16), you can use trinomial factoring techniques.
  • Quadratic Formula: If a polynomial simplifies to a quadratic equation (ax² + bx + c = 0), you can directly use the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a to find its roots. This is particularly useful when factoring is not straightforward.
• Graphical Tools/Technology 🖥️: Modern graphing tools like Desmos can visually determine x-intercepts. You input the polynomial equation, and the tool plots the graph, showing you where it crosses or touches the x-axis.
• Trial and Error / Long Division 🎲: For higher-degree polynomials where factoring isn’t obvious, you might guess some simple integer roots. If x = a is a root, then (x - a) is a factor. You can then use polynomial long division to divide the polynomial by this factor, reducing its degree and making it easier to find other roots.
• Intermediate Value Theorem (IVT) ↔️: This theorem applies to continuous functions (which polynomials are). It states that if a polynomial function f(x) takes on opposite signs at two points a and b (i.e., f(a) is positive and f(b) is negative, or vice-versa), then there must be at least one zero (an x-intercept) between a and b. This helps confirm the existence of a root in an interval even if you can’t find it precisely.

Characterising Zeros: Multiplicity 🔢

The multiplicity of a zero tells you how many times a particular factor (x - a) appears in the polynomial’s factored form. This ‘multiplicity’ is key to understanding how the graph behaves at each x-intercept.

• Even Multiplicity (e.g., (x-a)², (x-a)⁴): If a zero has an even multiplicity, the graph will touch the x-axis and “bounce off” it at that intercept, rather than crossing it.
  • As the even power increases (e.g., from x² to x⁴), the graph appears flatter as it approaches and leaves the zero.
• Odd Multiplicity (e.g., (x-a)¹, (x-a)³): If a zero has an odd multiplicity, the graph will cross or intersect the x-axis at that intercept.
  • If the graph appears almost linear at the intercept, it indicates a single (first) order multiplicity.
  • Higher odd powers (e.g., x⁵) will make the graph appear flatter while approaching and leaving the zero.
• Sum of Multiplicities ➕: The sum of the multiplicities of all real zeros (x-intercepts) should always be less than or equal to the degree of the polynomial. This is because polynomials can have complex roots which do not appear as x-intercepts on the real coordinate plane. For instance, a degree 4 polynomial might only have one visible x-intercept if the other two roots are complex.

By using these insights, you can not only find the zeros but also get a good sense of the polynomial’s overall shape and behaviour.

Practice Questions 🧠

Now, let’s put your understanding to the test!

Question 1: Finding Zeros by Factoring Find the x-intercepts (zeros) of the polynomial function: f(x) = x⁶ - 8x⁴ + 16x²

Question 2: Finding Zeros by Grouping Find the x-intercepts (zeros) of the polynomial function: f(x) = x³ - 4x² - 3x + 12

Question 3: Identifying Zeros and Multiplicities from Factored Form Given the polynomial function g(x) = (x-1)²(x+3). a) Identify all the x-intercepts. b) Determine the multiplicity of each x-intercept. c) Describe the behaviour of the graph at each x-intercept.

Question 4: Identifying Zeros and Multiplicities from a Graph Look at the graph of a degree 6 polynomial below (imagine a graph shown here as described in the source). The graph shows x-intercepts at x = -2, x = 0, and x = 2. a) Determine the multiplicity of the zero at x = -2 if the graph appears linear at this intercept. b) Determine the multiplicity of the zero at x = 0 if the graph shows an S-shape (appearing flattened as it crosses). c) Determine the multiplicity of the zero at x = 2 if the graph bounces off the x-axis. d) Verify if the sum of multiplicities equals the degree of the polynomial.

Solutions ✅

Solution 1: Given f(x) = x⁶ - 8x⁴ + 16x². To find the x-intercepts, set f(x) = 0: x⁶ - 8x⁴ + 16x² = 0

1. Factor out the Greatest Common Factor (GCF): The common monomial factor is x². x²(x⁴ - 8x² + 16) = 0

2. Factor the trinomial: The expression inside the parenthesis (x⁴ - 8x² + 16) resembles a quadratic equation if we let t = x² (so t² = x⁴). This would be t² - 8t + 16. This is a perfect square trinomial: (t - 4)². Substitute x² back for t: (x² - 4)². So the equation becomes: x²(x² - 4)² = 0.

3. Factor further: The term (x² - 4) is a difference of squares, which can be factored into (x - 2)(x + 2). So the equation is: x²((x - 2)(x + 2))² = 0. This can be rewritten as: x²(x - 2)²(x + 2)² = 0.

4. Set each factor to zero and solve:

   • x² = 0 ⇒ x = 0
   • (x - 2)² = 0 ⇒ x - 2 = 0 ⇒ x = 2
   • (x + 2)² = 0 ⇒ x + 2 = 0 ⇒ x = -2

The x-intercepts (zeros) are x = 0, x = 2, and x = -2.

Solution 2: Given f(x) = x³ - 4x² - 3x + 12. To find the x-intercepts, set f(x) = 0: x³ - 4x² - 3x + 12 = 0

1. Factor by Grouping: There is no common monomial for all terms. Group the terms in pairs. (x³ - 4x²) + (-3x + 12) = 0

2. Factor out GCF from each pair: x²(x - 4) - 3(x - 4) = 0

3. Factor out the common binomial: (x - 4) is common. (x² - 3)(x - 4) = 0

4. Set each factor to zero and solve:

   • x - 4 = 0 ⇒ x = 4
   • x² - 3 = 0 ⇒ x² = 3 ⇒ x = ±√3 (or x = ±1.732... approximately)

The x-intercepts (zeros) are x = 4, x = √3, and x = -√3.

Solution 3: Given g(x) = (x-1)²(x+3).

a) Identify all the x-intercepts: Set g(x) = 0.

• (x - 1)² = 0 ⇒ x - 1 = 0 ⇒ x = 1
• (x + 3) = 0 ⇒ x = -3

The x-intercepts are x = 1 and x = -3.

b) Determine the multiplicity of each x-intercept:

• For x = 1, the factor (x - 1) is raised to the power of 2. So, the multiplicity is 2 (an even multiplicity).
• For x = -3, the factor (x + 3) is raised to the power of 1 (implied). So, the multiplicity is 1 (an odd multiplicity).

c) Describe the behaviour of the graph at each x-intercept:

• At x = 1 (multiplicity 2, even): The graph will touch the x-axis and bounce off it at this point.
• At x = -3 (multiplicity 1, odd): The graph will cross the x-axis at this point, appearing almost linear.

Solution 4: Given a degree 6 polynomial with x-intercepts at x = -2, x = 0, and x = 2.

a) Multiplicity at x = -2: If the graph appears linear at x = -2, this indicates a multiplicity of 1 (single order). b) Multiplicity at x = 0: An S-shape (flattened as it crosses) at x = 0 indicates an odd multiplicity higher than 1. Given the degree is 6 and other multiplicities, this is likely 3 or 5. c) Multiplicity at x = 2: If the graph bounces off the x-axis at x = 2, this indicates an even multiplicity. Given the degree is 6, this could be 2 or 4.

d) Verify sum of multiplicities: The sum of all multiplicities must be less than or equal to the degree of the polynomial, which is 6.

• Assume multiplicity at x = -2 is 1.
• Assume multiplicity at x = 0 is 3 (S-shape, odd).
• Assume multiplicity at x = 2 is 2 (bounces off, even).

Sum = 1 (at x=-2) + 3 (at x=0) + 2 (at x=2) = 6. This sum equals the degree of the polynomial, so these multiplicities are consistent with the graph of a degree 6 polynomial.Understanding the zeros of a polynomial function is like finding the special points where the graph of the function crosses or touches the horizontal axis (the x-axis) 🎯. These points are also known as x-intercepts. For a polynomial function, f(x), a value of x is a zero if f(x) = 0.

Let’s break down how to identify and characterise these zeros from a graph or an equation:

Identifying Zeros of Polynomials 🕵️‍♀️

When you’re trying to find the zeros of a polynomial, you’re essentially looking for the x-values that make the function’s output zero. Here’s how you can find them:

• Factoring Technique ✂️: This is a crucial technique for finding zeros. If you can factor the polynomial function’s equation into a product of simpler expressions, you can then set each factor equal to zero and solve for x.
  • Greatest Common Factor (GCF): Sometimes, all terms in a polynomial share a common monomial factor, which can be factored out first.
  • Factoring by Grouping: For polynomials with four or more terms, you might group terms to find common factors.
  • Trinomial Factoring: For expressions resembling quadratic equations (e.g., ax² + bx + c or forms like t² - 8t + 16), you can use trinomial factoring techniques.
  • Quadratic Formula: If a polynomial simplifies to a quadratic equation (ax² + bx + c = 0), you can directly use the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a to find its roots. This is particularly useful when factoring is not straightforward.
• Graphical Tools/Technology 🖥️: Modern graphing tools like Desmos can visually determine x-intercepts. You input the polynomial equation, and the tool plots the graph, showing you where it crosses or touches the x-axis.
• Trial and Error / Long Division 🎲: For higher-degree polynomials where factoring isn’t obvious, you might guess some simple integer roots. If x = a is a root, then (x - a) is a factor. You can then use polynomial long division to divide the polynomial by this factor, reducing its degree and making it easier to find other roots.
• Intermediate Value Theorem (IVT) ↔️: This theorem applies to continuous functions (which polynomials are). It states that if a polynomial function f(x) takes on opposite signs at two points a and b (i.e., f(a) is positive and f(b) is negative, or vice-versa), then there must be at least one zero (an x-intercept) between a and b. This helps confirm the existence of a root in an interval even if you can’t find it precisely.

Characterising Zeros: Multiplicity 🔢

The multiplicity of a zero tells you how many times a particular factor (x - a) appears in the polynomial’s factored form. This ‘multiplicity’ is key to understanding how the graph behaves at each x-intercept.

• Even Multiplicity (e.g., (x-a)², (x-a)⁴): If a zero has an even multiplicity, the graph will touch the x-axis and “bounce off” it at that intercept, rather than crossing it.
  • As the even power increases (e.g., from x² to x⁴), the graph appears flatter as it approaches and leaves the zero.
• Odd Multiplicity (e.g., (x-a)¹, (x-a)³): If a zero has an odd multiplicity, the graph will cross or intersect the x-axis at that intercept.
  • If the graph appears almost linear at the intercept, it indicates a single (first) order multiplicity.
  • Higher odd powers (e.g., x⁵) will make the graph appear flatter while approaching and leaving the zero.
• Sum of Multiplicities ➕: The sum of the multiplicities of all real zeros (x-intercepts) should always be less than or equal to the degree of the polynomial. This is because polynomials can have complex roots which do not appear as x-intercepts on the real coordinate plane. For instance, a degree 4 polynomial might only have one visible x-intercept if the other two roots are complex.

By using these insights, you can not only find the zeros but also get a good sense of the polynomial’s overall shape and behaviour.

Practice Questions 🧠

Now, let’s put your understanding to the test!

Question 1: Finding Zeros by Factoring Find the x-intercepts (zeros) of the polynomial function: f(x) = x⁶ - 8x⁴ + 16x²

Question 2: Finding Zeros by Grouping Find the x-intercepts (zeros) of the polynomial function: f(x) = x³ - 4x² - 3x + 12

Question 3: Identifying Zeros and Multiplicities from Factored Form Given the polynomial function g(x) = (x-1)²(x+3). a) Identify all the x-intercepts. b) Determine the multiplicity of each x-intercept. c) Describe the behaviour of the graph at each x-intercept.

Question 4: Identifying Zeros and Multiplicities from a Graph Imagine a graph of a degree 6 polynomial where:

• At x = -2, the graph appears linear as it crosses the x-axis.
• At x = 0, the graph shows an S-shape (flattened as it crosses) through the x-axis.
• At x = 2, the graph bounces off the x-axis.

a) Determine the multiplicity of the zero at x = -2. b) Determine the multiplicity of the zero at x = 0. c) Determine the multiplicity of the zero at x = 2. d) Verify if the sum of multiplicities equals the degree of the polynomial.

Solutions ✅

Solution 1: Given f(x) = x⁶ - 8x⁴ + 16x². To find the x-intercepts, set f(x) = 0: x⁶ - 8x⁴ + 16x² = 0

1. Factor out the Greatest Common Factor (GCF): The common monomial factor is x². x²(x⁴ - 8x² + 16) = 0

2. Factor the trinomial: The expression inside the parenthesis (x⁴ - 8x² + 16) resembles a quadratic equation if we let t = x² (so t² = x⁴). This would be t² - 8t + 16. This is a perfect square trinomial: (t - 4)². Substitute x² back for t: (x² - 4)². So the equation becomes: x²(x² - 4)² = 0.

3. Factor further: The term (x² - 4) is a difference of squares, which can be factored into (x - 2)(x + 2). So the equation is: x²((x - 2)(x + 2))² = 0. This can be rewritten as: x²(x - 2)²(x + 2)² = 0.

4. Set each factor to zero and solve:

   • x² = 0 ⇒ x = 0
   • (x - 2)² = 0 ⇒ x - 2 = 0 ⇒ x = 2
   • (x + 2)² = 0 ⇒ x + 2 = 0 ⇒ x = -2

The x-intercepts (zeros) are x = 0, x = 2, and x = -2.

Solution 2: Given f(x) = x³ - 4x² - 3x + 12. To find the x-intercepts, set f(x) = 0: x³ - 4x² - 3x + 12 = 0

1. Factor by Grouping: There is no common monomial for all terms. Group the terms in pairs. (x³ - 4x²) + (-3x + 12) = 0

2. Factor out GCF from each pair: x²(x - 4) - 3(x - 4) = 0

3. Factor out the common binomial: (x - 4) is common. (x² - 3)(x - 4) = 0

4. Set each factor to zero and solve:

   • x - 4 = 0 ⇒ x = 4
   • x² - 3 = 0 ⇒ x² = 3 ⇒ x = ±√3 (or x = ±1.732... approximately)

The x-intercepts (zeros) are x = 4, x = √3, and x = -√3.

Solution 3: Given g(x) = (x-1)²(x+3).

a) Identify all the x-intercepts: Set g(x) = 0.

• (x - 1)² = 0 ⇒ x - 1 = 0 ⇒ x = 1
• (x + 3) = 0 ⇒ x = -3

The x-intercepts are x = 1 and x = -3.

b) Determine the multiplicity of each x-intercept:

• For x = 1, the factor (x - 1) is raised to the power of 2. So, the multiplicity is 2 (an even multiplicity).
• For x = -3, the factor (x + 3) is raised to the power of 1 (implied). So, the multiplicity is 1 (an odd multiplicity).

c) Describe the behaviour of the graph at each x-intercept:

• At x = 1 (multiplicity 2, even): The graph will touch the x-axis and bounce off it at this point.
• At x = -3 (multiplicity 1, odd): The graph will cross the x-axis at this point, appearing almost linear.

Solution 4: Given a degree 6 polynomial with x-intercepts at x = -2, x = 0, and x = 2.

a) Multiplicity at x = -2: If the graph appears linear at x = -2, this indicates a multiplicity of 1 (single order). b) Multiplicity at x = 0: An S-shape (flattened as it crosses) at x = 0 indicates an odd multiplicity higher than 1. Given the degree is 6 and other multiplicities, this is determined to be 3. c) Multiplicity at x = 2: If the graph bounces off the x-axis at x = 2, this indicates an even multiplicity. Given the degree is 6, this is determined to be 2.

d) Verify sum of multiplicities: The sum of all multiplicities must be less than or equal to the degree of the polynomial, which is 6.

• Multiplicity at x = -2: 1
• Multiplicity at x = 0: 3
• Multiplicity at x = 2: 2

Sum = 1 + 3 + 2 = 6. This sum equals the degree of the polynomial, so these multiplicities are consistent with the graph of a degree 6 polynomial.