Mathematics
Grade 11
15 min
Find the roots of factored polynomials
Find the roots of factored polynomials
What you'll learn
- Identify the product of any two numbers from 1 to 10 with 80% accuracy on a written quiz.
- Solve multiplication problems from 1x1 to 10x10 by using manipulatives (like counters or drawings) to show groups of objects and find the total, with at least 7 out of 10 problems solved correctly.
- Explain the meaning of multiplication as repeated addition, using examples from the multiplication tables up to 10, in their own words.
- Apply knowledge of multiplication facts up to 10 to solve simple word problems involving equal groups (e.g., 'If each of 6 friends has 3 cookies, how many cookies are there in total?') with 75% accuracy.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a root (or zero) of a polynomial and its relationship to the x-intercepts of its graph.
Apply the Zero Product Property to find the real roots of a polynomial given in factored form.
Determine the multiplicity of each root from the exponent of its corresponding factor.
Solve for complex roots from irreducible quadratic factors using the quadratic formula or by isolating the variable.
Relate the degree of the polynomial to the total number of roots, including multiplicities and complex roots, as stated by the Fundamental Theorem of Algebra.
Differentiate between real and complex roots and their graphical representations.
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Key Concepts & Vocabulary
TermDefinitionExample
Root (or Zero)A value of the variable (e.g., x) that makes a polynomial equal to zero. The real roots of a polynomial correspond to the x-intercepts of its graph.For the polynomial P(x) = x - 5, the root is x = 5 because P(5) = 5 - 5 = 0.
Factored FormA polynomial expressed as a product of its factors, which are typically simpler polynomials like linear or quadratic expressions.P(x) = x^2 + x - 6 can be written in factored form as P(x) = (x + 3)(x - 2).
Zero Product PropertyA fundamental rule stating that if the product of two or more factors is zero, then at least one of the factors must be zero.If (x - 2)(x + 3) = 0, then either x - 2 = 0 or x + 3 = 0.
MultiplicityThe number of times a particular root occurs, which is determined by the exponent on its corresponding...
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Core Formulas
Zero Product Property
If P(x) = f_1(x) \cdot f_2(x) \cdot ... \cdot f_n(x) = 0, then f_1(x) = 0 or f_2(x) = 0 or ... or f_n(x) = 0.
To find the roots of a polynomial in factored form, set each individual factor equal to zero and solve the resulting simpler equations for the variable.
Quadratic Formula
For a quadratic factor ax^2 + bx + c = 0, the roots are given by x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Use this formula to find the roots of any quadratic factor that cannot be easily factored further. It is essential for finding complex roots, which occur when the discriminant (b^2 - 4ac) is negative.
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Challenging
Which of the following polynomials in factored form has roots x = -1 (with multiplicity 2) and the complex conjugate pair x = 2 ± i?
A.P(x) = (x + 1)^2(x^2 - 4x + 5)
B.P(x) = (x - 1)^2(x^2 - 4x + 3)
C.P(x) = (x + 1)^2(x^2 + 4x + 5)
D.P(x) = (x - 1)^2(x^2 + 4x - 5)
Challenging
The polynomial P(x) = (x^2 - k)(x^2 + 4x + 5) has a root at x = 3. What are the other three roots?
A.x = -3, x = -2 + i, x = -2 - i
B.x = 3, x = -2 + i, x = -2 - i
C.x = -3, x = -2 + i, x = -2 - i
D.x = -3, x = 2 + i, x = 2 - i
Challenging
Consider the polynomial P(x) = (x^2 + 9)^2(x - 1)^3. Which statement is correct?
A.The polynomial has degree 5 and 3 real roots.
B.The polynomial has 4 roots in total.
C.The polynomial has a real root x = 1 with multiplicity 2.
D.The polynomial has degree 7 and complex roots 3i and -3i, each with multiplicity 2.
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What grade level is "Find the roots of factored polynomials"?
Find the roots of factored polynomials is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Find the roots of factored polynomials?
You'll be able to: Identify the product of any two numbers from 1 to 10 with 80% accuracy on a written quiz; Solve multiplication problems from 1x1 to 10x10 by using manipulatives (like counters or drawings) to show groups of objects and find the….
Is "Find the roots of factored polynomials" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Find the roots of factored polynomials?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.