## 17Calculus Precalculus - Build The Polynomial Given The Roots

##### 17Calculus

Find The Polynomial Given The Roots

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.

### MIP4U - Zeros, Factors and Graphs of Polynomial Functions [10min-2secs]

video by MIP4U

Okay, time for you to try your hand at the practice problems.

Practice

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

### PatrickJMT - 1637 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

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

### MIP4U - 1643 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Find a polynomial function with zeros $$-6$$ and $$i$$.

Problem Statement

Find a polynomial function with real coefficients having zeros $$-6$$ and $$i$$.

Solution

### MIP4U - 1645 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

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

### MIP4U - 1656 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Find a polynomial with roots $$-4, -1, 3$$.

Problem Statement

Find a polynomial with roots $$-4, -1, 3$$.

Solution

### MIP4U - 1659 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

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

### MIP4U - 1654 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

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

### MIP4U - 1649 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

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

### PatrickJMT - 1639 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

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

### PatrickJMT - 1640 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

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

### MIP4U - 1657 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Find a polynomial with zeros $$3, -1, 4+2i$$.

Problem Statement

Find a polynomial with real coefficients and zeros $$3, -1, 4+2i$$.

Solution

### MIP4U - 1653 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

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

### MIP4U - 1658 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.