# Precalculus : Solving Polynomial Equations

## Example Questions

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### Example Question #1 : Solving Polynomial Equations

A polynomial with leading term has roots 1,2, and 3. What is the polynomial?

Explanation:

Given the roots of a polynomial, the problem can be solved in reverse. For 3,2, and 1 to be roots, the following must be true:

Therefore, expand the left side of the equation to find the polynomial.

### Example Question #2 : Solving Polynomial Equations

Solve for x:

Explanation:

To solve, we must expand, factor, and solve.

### Example Question #3 : Solving Polynomial Equations

Solve the following for x:

Explanation:

To solve, we must expand, factor, and solve for x.

### Example Question #4 : Solving Polynomial Equations

Given the polynomial , determine the value of

given that the polynomial is exactly divisable by

Explanation:

First determine the value of x by setting  equal to 0.

Next, plug in the value of x determined above into the polynomial. The polynomial is to be set equal to 0. Solve for the value of h.

### Example Question #5 : Solving Polynomial Equations

Solve the following for x:

Explanation:

To solve, we must expand, factor, and solve for x.

### Example Question #1 : Solve A Polynomial By Factoring

Solve the following polynomial for  by factoring:

Explanation:

The polynomial in the problem is given as follows:

Factoring this polynomial, we would get an expression of the form:

So we need to determine what a and b are. We know we need two factors that when multiplied equal -12, and when added equal -1. If we consider 2 and 6, we could get -12 but could not arrange them in any way that would make their sum equal to -1. We then look at 3 and 4, whose product can be -12 is one of them is negative, and whose sum can be -1 if -4 is added to 3. This tells us that the 4 must be the negative factor and the 3 must be the positive factor, so we get the following:

### Example Question #2 : Solve A Polynomial By Factoring

Solve the following polynomial for  by factoring:

Explanation:

Factoring our polynomial, we can see we will have 2x and x at the beginning of each factor, while we need to find two numbers whose product is 15 and whose sum when multiplied by our leading terms and added is -11x. This gives us the following factorization:

So now we can set each term equal to 0 and solve for our two values of x:

### Example Question #1 : Solve A Polynomial By Factoring

Find the root(s) for the function:

Explanation:

The function  is in the form , and can be factorized.

Determine the values of  and .

Substitute this into the formula.

Set  to find all roots.

### Example Question #1 : Solve A Polynomial By Factoring

Solve the following equation:

Explanation:

To solve, factor and then solve for x.

In order to factor we need all our variables and constants on one side. Add 18 to both sides to make our function in a form for which we can factor.

Now we want to find factors of 18 that when added together give us 9. Thus we get the following factored form.

Now set each factor equal to zero and solve for x.

### Example Question #5 : Solve A Polynomial By Factoring

Find the zeros of the function .

Explanation:

To find the zeros of the function, you need to factor the equation. Using trial and error, you should arrive at: . Then set those expressions equal to  so that your roots are .

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