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Coefficients and Roots

Reference > Mathematics > Algebra > Polynomials
 

Often the words "zeroes" and "roots" will be used interchangeably. However, they are not the same thing, and it is good to understand the difference.

A zero is a value for which a polynomial is equal to zero.

A root is a value for which a polynomial equation is true.

Why are these terms used interchangeably sometimes? Because if you set a polynomial equal to zero, you have a polynomial equation, and that equation's roots are the same as the polynomial's zeroes.

For example, the polynomial x - 10 has one zero: x = 10. And the polynomial equation x - 10 = 0 has one root: x = 10.

In this section we're going to deal strictly with polynomial equations in which a polynomial is set equal to zero. We're going to begin by looking at some quadratic equations, and explore the relationship between the coefficients and the roots. From there, we will expand the concept to higher degree polynomials.

Let's consider the quadratic equation x- 7x + 12 = 0. The roots of this equation (which you can find by factoring) are 3 and 4. Isn't it interesting that the roots multiply to 12 (which is the constant term in the equation) and add to 7 (which is the opposite of the coefficient of x)! We should wonder if that's always the case. Let's try another example: x2 + 14x + 48 = 0. In this case, the roots are -6 and -8. Sure enough, they add to the opposite of 14, and multiply to 48.

But let's not be hasty to draw a conclusion; let's try this equation: 2x2 - x - 15 = 0. This one has roots x = -5/2 and x = 3. These add to 1/2 and multiply to 15/2. This doesn't quite match what we saw before. However, if we take the coefficient of x and divide it by the coefficient of x2, we get -1/2, and if we divide the constant term by the coefficient of x2, we get -15/2. and these two numbers match our pattern.

It turns out that this is a rule that is always true: the sum of the roots of a quadratic is the opposite of the coefficient of x divided by the leading coefficient, and the product of the roots is always the constant term divided by the leading coefficient.

To put it another way: In the quadratic equation ax2 + bx + c = 0, the roots add to -b/a, and they multiply to c/a.

Example: Find a quadratic which has 5 and 7 as its roots.

Solution: 5 + 7 = 12, and 5 x 7 = 35, so a quadratic equation could be x2 - 12x + 35 = 0

Note that in this question, I asked you to find A quadratic, not to find THE quadratic. This is because there are multiple quadratic equations that match the requirements. Consider what happens if you multiply each term by 2: 2x2 - 24x + 70 = 0. Does it have the same roots? Yes it does! So it's good to remember that we can have multiple solutions. Generally we give the quadratic in which all the terms are relatively prime (in the case of my second answer, all the coefficients are divisible by 2, so they're not relatively prime).

Example: Find b if one of the roots of 2x2 + bx + 12 = 0 is 3.

Solution: The product of the roots is 12/2 = 6. Since one of the roots is 3, the other must be 2. The sum of the roots is therefore 5, and thus -b/2 = 5, or b = -10.

Example: The roots of x2 - 7x + c = 0 are two integers which differ by 3. Find c.

Solution: The sum of the roots is 7. If the roots are m and n, then m + n = 7 and m - n = 3, which leads to m = 5, n = 2. Thus, since c is the product of the roots, c = 10.

Questions

1.
A quadratic equation has roots 3 and 7. Give a possible equation that matches these criteria.
2.
A quadratic equation has roots 4 and m. The coefficient of x2 is 2 and the coefficient of x is 18. Find m.
3.
What is the sum of the roots of the quadratic 4x2 - 10x + 5 = 0?
4.
What is the product of the roots of 3x2 + 5x - 18 = 0
5.
What is the product of the roots of 2x2 + 8x = -20
6.
The roots of x2 - 11x + k = 0 are consecutive integers. Find k.
7.
One of the roots of a quadratic equation is twice the other. The constant term and the coefficient of x are opposites of each other, but are not equal to zero. What are the roots?
8.
The product of the roots of a quadratic is 12 more than the smaller root. Find k if the quadratic is x2 + kx +15 = 0
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Fundamental Theorem of AlgebraFundamental Theorem of Algebra
Higher Degree Coefficients and RootsHigher Degree Coefficients and Roots
 

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