Unit 2: Factoring Higher-Degree Polynomials and Zeros
Cracking degree-three-and-up polynomials into their pieces, and reading what each piece says about the graph's roots.
7 topics · 25 questions
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Factoring by Grouping
- How it worksHow do you group four terms into two pairs that each share a factor?
- Worked exampleFind the complete factorization of x3 + 3x2 - 4x - 12 using grouping.
- Worked exampleFind the complete factorization of 2x3 + 6x2 - 5x - 15 by grouping the terms into pairs.
Your turn1 of 2
Find the complete factorization of x2(x + 7) - 7(x + 7).Sum and Difference of Cubes
- How it worksHow does the sum-of-cubes pattern turn into a linear factor times a quadratic factor?
- Worked exampleFind the complete factorization of 8x3 - 27 using the difference of cubes pattern.
- Worked exampleFind the complete factorization of x3 + 125 using the sum of cubes pattern.
Your turn1 of 2
Find the complete factorization of (5x)3 - 73.Factoring Higher-Degree Polynomials
- How it worksHow does substituting u = x2 turn a quartic into a quadratic you already know how to factor?
- Worked exampleFind the complete factorization of x4 - 5x2 + 4 using substitution.
- Common mistakeWhy doesn't factoring out the greatest common factor first mean the polynomial is fully factored?
- Going furtherWhat happens when the substitution trick is tried on a polynomial like x4 + x2 + 1 instead?
Your turn1 of 2
Let u = x2. Rewrite x4 - 7x2 + 6 in terms of u.The Rational Root Theorem
- How it worksHow do the constant term and leading coefficient generate the list of candidate rational roots?
- Worked exampleFind every possible rational root of 2x3 - 3x2 - 8x - 3 and identify the one that actually works.
- Common mistakeWhy doesn't the rational root theorem guarantee that a polynomial has any rational roots at all?
Your turn1 of 2
A cubic polynomial has leading coefficient 4 and constant term 8 (both positive integers). According to the rational root theorem, which of these IS a valid candidate rational root?Finding All Zeros of a Polynomial
- How it worksWalk me through finding every zero of a cubic by combining synthetic division with factoring.
- Worked exampleFind all zeros of x3 - 4x2 + x + 6.
- Common mistakeWhy can a degree-three polynomial have fewer than three distinct real zeros?
- Going furtherWhat changes about the zero-finding process when one of a cubic's real zeros has multiplicity two?
Your turn1 of 2
A cubic polynomial x3 + 7x2 + qx + r (for some q and r) has x = 5 as one of its zeros. Find the sum of its other two zeros.Multiplicity and the Bounce-or-Cross Rule
- How it worksHow does the power on a repeated factor decide whether the graph bounces or crosses there?
- Worked exampleSketch how y = (x - 1)2(x + 3) behaves at each of its two x-intercepts.
- Common mistakeWhy doesn't a factor's multiplicity being even always mean the root is a minimum point?
- Going furtherWhat happens to the graph right at a root with multiplicity three instead of one or two?
Your turn1 of 2
A polynomial's factorization includes (x - 7)2 as a factor, and no other factor is zero at x = 7. What does the graph do at x = 7?The Fundamental Theorem of Algebra
- How it worksHow does the fundamental theorem of algebra guarantee exactly n roots for a degree-n polynomial once you count complex ones?
- Worked exampleFind how many total roots x5 - 3x3 + x - 1 has, counting multiplicity, using the fundamental theorem of algebra.
- Common mistakeWhy doesn't a polynomial with no real roots visible on its graph violate the fundamental theorem of algebra?
- Ties togetherHow do complex conjugate pairs of roots connect the fundamental theorem of algebra to complex numbers?
Your turn1 of 2
According to the fundamental theorem of algebra, how many total roots (counting multiplicity) does a degree-9 polynomial have?