Find The Polynomial Given The Roots
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From the discussion on the polynomial zeroes/roots page, finding the roots is pretty straightforward as long as you can factor the polynomial. Another thing you will be asked to do is find the polynomial given the roots. This, also, is pretty straightforward if you remember just a couple of things.
1. In precalculus you will almost certainly always be working with polynomials with real coefficients. This is important since the next point follows from and is dependent on this fact.
2. For complex roots of polynomials with real coefficients, the complex roots always appear in complex conjugate pairs. For example, if one of your roots is \(2+3i\), the other root is guaranteed to be \(2-3i\).
3. Another situation to watch for is something called multiplicity. The root can appear more than once and, therefore, have a multiplicity greater than one. For example, the roots of \(x^2-6x+9=(x-3)^2\) are both \(x=3\), so we say that the roots are 3 with multiplicity 2 (since it appears two times). Multiplicity just tells us how many times the root exists.
To solve a problem where you are given the roots and you need to come up with the polynomial, you just write the factors and multiply out. Also, most instructors will want the resulting polynomial to have integer coefficients. For example, if you multiply out and get a polynomial like \((1/2)x^2+3\), multiply by 2 to get \(x^3+6\). Notice this polynomial has the same roots as the first one even though it is not the same polynomial. Of course, it depends on the problem statement, so check with your instructor to see what they require.
Before jumping into the practice problems, let's watch a quick video explaining these techniques in more detail and how the roots look on a graph. This video also includes lots of examples.
video by MIP4U |
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Okay, time for you to try your hand at the practice problems.
Practice
Solve these problems giving your answers in exact form. Make sure that all of the polynomials in your answers have real coefficients.
Find the polynomial \(f(x)\) of degree 3 with zeros \(-1, ~2, ~4\) where \(f(1)=8\).
Problem Statement
Find the polynomial \(f(x)\) of degree 3 with zeros \(-1, ~2, ~4\) where \(f(1)=8\).
Solution
video by PatrickJMT |
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Find a polynomial function with zeros \(4\) and \(2+3i\).
Problem Statement
Find a polynomial function with real coefficients having zeros \(4\) and \(2+3i\).
Solution
video by MIP4U |
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Find a polynomial function with zeros \(-6\) and \(i\).
Problem Statement
Find a polynomial function with real coefficients having zeros \(-6\) and \(i\).
Solution
video by MIP4U |
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Find a polynomial with roots \(-2/3, 3/7, -1\).
Problem Statement
Find a polynomial with real coefficients having roots \(-2/3, 3/7, -1\).
Solution
video by MIP4U |
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Find a polynomial with roots \(-4, -1, 3\).
Problem Statement
Find a polynomial with roots \(-4, -1, 3\).
Solution
video by MIP4U |
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Find a polynomial of degree 3 with roots \(x=-2\) (multiplicity 2) and \(x=3\) through the point \((2,80)\).
Problem Statement
Find a polynomial with real coefficients of degree 3 with roots \(x=-2\) (multiplicity 2) and \(x=3\) through the point \((2,80)\).
Solution
video by MIP4U |
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Find the real polynomial of degree 4 having roots \(x=4\) (multiplicity 2), \(x=-3\) and \(x=0\) passing through the point \((-2,-36)\).
Problem Statement
Find the real polynomial of degree 4 having roots \(x=4\) (multiplicity 2), \(x=-3\) and \(x=0\) passing through the point \((-2,-36)\).
Solution
video by MIP4U |
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Find the polynomial \(f(x)\) of degree 3 where two roots are \(x=1\) and \(x=-2\), the leading coefficient is \(-1\) and \(f(3)=48\).
Problem Statement
Find the polynomial \(f(x)\) of degree 3 where two roots are \(x=1\) and \(x=-2\), the leading coefficient is \(-1\) and \(f(3)=48\).
Solution
video by PatrickJMT |
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Write an expression for a polynomial \(f(x)\) of degree 3 with zeros \(x=2\) and \(x=-2\), leading coefficient of 1 and \(f(-4)=30\).
Problem Statement
Write an expression for a polynomial \(f(x)\) of degree 3 with zeros \(x=2\) and \(x=-2\), leading coefficient of 1 and \(f(-4)=30\).
Solution
video by PatrickJMT |
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Find a polynomial with roots \(x=2\) (multiplicity 2) and \(x=-1-2i\).
Problem Statement
Find a polynomial with roots \(x=2\) (multiplicity 2) and \(x=-1-2i\).
Solution
video by MIP4U |
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Find a polynomial with zeros \(3, -1, 4+2i\).
Problem Statement
Find a polynomial with real coefficients and zeros \(3, -1, 4+2i\).
Solution
video by MIP4U |
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Find a polynomial with roots \( 1/4, -2/3, -2-i\).
Problem Statement
Find a polynomial with roots \( 1/4, -2/3, -2-i\).
Solution
video by MIP4U |
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